state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : NeBot l
h₁ : Tendsto (fun x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using h₂.mul h₁ | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr := simps)
"Construct an additive unit from limits of additive units and their negatives."]
def Filter.Tendsto.units [TopologicalSpace N] [Monoid N] [ContinuousMul N] [T2Space N]
{f : ι → Nˣ} {r₁ r₂ : N} {l : Filter ι} [l.NeBot] (... | Mathlib.Topology.Algebra.Monoid.197_0.3p9EZf9ZWFxWOAq | /-- Construct a unit from limits of units and their inverses. -/
@[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [continuous_iff_continuousAt] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rintro ⟨x₀, y₀⟩ | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | have key : (fun p : M × M => x₀ * p.1 * (p.2 * y₀)) =
((fun x => x₀ * x) ∘ fun x => x * y₀) ∘ uncurry (· * ·) := by
ext p
simp [uncurry, mul_assoc] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | ext p | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
case h
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp [uncurry, mul_assoc] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | have key₂ : ((fun x => x₀ * x) ∘ fun x => y₀ * x) = fun x => x₀ * y₀ * x := by
ext x
simp [mul_assoc] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | ext x | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
case h
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp [mul_assoc] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | calc
map (uncurry (· * ·)) (𝓝 (x₀, y₀)) = map (uncurry (· * ·)) (𝓝 x₀ ×ˢ 𝓝 y₀) :=
by rw [nhds_prod_eq]
_ = map (fun p : M × M => x₀ * p.1 * (p.2 * y₀)) (𝓝 1 ×ˢ 𝓝 1) := by
-- Porting note: `rw` was able to prove this
-- Now it fails with `failed to rewrite using equation theorems... | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [nhds_prod_eq] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp (config := { unfoldPartialApp := true }) only [uncurry] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp_rw [hleft x₀, hright y₀, prod_map_map_eq, Filter.map_map, Function.comp_def] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [key, ← Filter.map_map] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : Monoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ = map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [← Filter.map_map, ← hright, hleft y₀, Filter.map_map, key₂, ← hleft] | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M :=
⟨by
... | Mathlib.Topology.Algebra.Monoid.263_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem ContinuousMul.of_nhds_one {M : Type u} [Monoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) <| 𝓝 1)
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ =... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | apply ContinuousMul.of_nhds_one hmul hleft | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M := by
| Mathlib.Topology.Algebra.Monoid.296_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ =... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | intro x₀ | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M := by
apply ContinuousMul.of_nhds_one hmul hleft
| Mathlib.Topology.Algebra.Monoid.296_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace M✝
inst✝³ : Mul M✝
inst✝² : ContinuousMul M✝
M : Type u
inst✝¹ : CommMonoid M
inst✝ : TopologicalSpace M
hmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : M), 𝓝 x₀ =... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp_rw [mul_comm, hleft x₀] | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M := by
apply ContinuousMul.of_nhds_one hmul hleft
intro x... | Mathlib.Topology.Algebra.Monoid.296_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
M₁ : Type u_6
M₂ : Type u_7
inst✝⁴ : TopologicalSpace M₂
inst✝³ : T2Space M₂
inst✝² : Mul M₁
inst✝¹ : Mul M₂
inst✝ : ContinuousMul M₂
⊢ IsClosed {f | ∀ (x y : M₁), f (x * y) = f x * f y} | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [setOf_forall] | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } := by
| Mathlib.Topology.Algebra.Monoid.318_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
M₁ : Type u_6
M₂ : Type u_7
inst✝⁴ : TopologicalSpace M₂
inst✝³ : T2Space M₂
inst✝² : Mul M₁
inst✝¹ : Mul M₂
inst✝ : ContinuousMul M₂
⊢ IsClosed (⋂ i, ⋂ i_1, {x | x (i * i_1) = x i * x i_1}) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (continuous_apply _)
-- Porting note: proof was:
-- `((continuous_apply _).mul (continuous_apply _))`
(by continuity) | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } := by
simp only [setOf_forall]
| Mathlib.Topology.Algebra.Monoid.318_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁵ : TopologicalSpace X
M₁ : Type u_6
M₂ : Type u_7
inst✝⁴ : TopologicalSpace M₂
inst✝³ : T2Space M₂
inst✝² : Mul M₁
inst✝¹ : Mul M₂
inst✝ : ContinuousMul M₂
x y : M₁
⊢ Continuous fun x_1 => x_1 x * x_1 y | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | continuity | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } := by
simp only [setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (continuous_apply _)
-- Porting note: proof ... | Mathlib.Topology.Algebra.Monoid.318_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem isClosed_setOf_map_mul [Mul M₁] [Mul M₂] [ContinuousMul M₂] :
IsClosed { f : M₁ → M₂ | ∀ x y, f (x * y) = f x * f y } | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N✝ : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace X
M : Type u_6
N : Type u_7
F : Type u_8
inst✝⁵ : Mul M
inst✝⁴ : Mul N
inst✝³ : MulHomClass F M N
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace N
inst✝ : ContinuousMul N
f : F
hf : Inducing ⇑f
⊢ Continuous (⇑f ∘ fun p =... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa only [(· ∘ ·), map_mul f] using hf.continuous.fst'.mul hf.continuous.snd' | @[to_additive]
theorem Inducing.continuousMul {M N F : Type*} [Mul M] [Mul N] [MulHomClass F M N]
[TopologicalSpace M] [TopologicalSpace N] [ContinuousMul N] (f : F) (hf : Inducing f) :
ContinuousMul M :=
⟨hf.continuous_iff.2 <| by
| Mathlib.Topology.Algebra.Monoid.375_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem Inducing.continuousMul {M N F : Type*} [Mul M] [Mul N] [MulHomClass F M N]
[TopologicalSpace M] [TopologicalSpace N] [ContinuousMul N] (f : F) (hf : Inducing f) :
ContinuousMul M | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
s : Set M
hs : s ∈ 𝓝 1
⊢ ∃ V, IsOpen V ∧ 1 ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | have : (fun a : M × M => a.1 * a.2) ⁻¹' s ∈ 𝓝 ((1, 1) : M × M) :=
tendsto_mul (by simpa only [one_mul] using hs) | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s := by
| Mathlib.Topology.Algebra.Monoid.475_0.3p9EZf9ZWFxWOAq | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
s : Set M
hs : s ∈ 𝓝 1
⊢ s ∈ 𝓝 (1 * 1) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa only [one_mul] using hs | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s := by
have : (fun a : M × M => a.1 * a.2) ⁻¹' s ∈ 𝓝 ((1, 1) : M × M) :=
tendsto_mul (by | Mathlib.Topology.Algebra.Monoid.475_0.3p9EZf9ZWFxWOAq | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
s : Set M
hs : s ∈ 𝓝 1
this : (fun a => a.1 * a.2) ⁻¹' s ∈ 𝓝 (1, 1)
⊢ ∃ V, IsOpen V ∧ 1 ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa only [prod_subset_iff] using exists_nhds_square this | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s := by
have : (fun a : M × M => a.1 * a.2) ⁻¹' s ∈ 𝓝 ((1, 1) : M × M) :=
tendsto_mul (by simpa only [one_mul] using hs)
| Mathlib.Topology.Algebra.Monoid.475_0.3p9EZf9ZWFxWOAq | @[to_additive exists_open_nhds_zero_half]
theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
⊢ ∃ V ∈ 𝓝 1, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases exists_nhds_one_split hu with ⟨W, W1, h⟩ | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
| Mathlib.Topology.Algebra.Monoid.492_0.3p9EZf9ZWFxWOAq | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | Mathlib_Topology_Algebra_Monoid |
case intro.intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
⊢ ∃ V ∈ 𝓝 1, ∀ {v w s t : M}, v ∈ V → w ∈ V → s ∈ V → t ∈ V... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩ | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩
| Mathlib.Topology.Algebra.Monoid.492_0.3p9EZf9ZWFxWOAq | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | Mathlib_Topology_Algebra_Monoid |
case intro.intro.intro.intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
V : Set M
V1 : V ∈ 𝓝 1
h' : ∀ v ∈ V, ∀ w ∈ V, v... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | use V, V1 | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩
rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩
| Mathlib.Topology.Algebra.Monoid.492_0.3p9EZf9ZWFxWOAq | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | Mathlib_Topology_Algebra_Monoid |
case right
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
V : Set M
V1 : V ∈ 𝓝 1
h' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W
⊢ ∀ {v w ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | intro v w s t v_in w_in s_in t_in | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩
rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩
use V, V1
| Mathlib.Topology.Algebra.Monoid.492_0.3p9EZf9ZWFxWOAq | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | Mathlib_Topology_Algebra_Monoid |
case right
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
u : Set M
hu : u ∈ 𝓝 1
W : Set M
W1 : W ∈ 𝓝 1
h : ∀ v ∈ W, ∀ w ∈ W, v * w ∈ u
V : Set M
V1 : V ∈ 𝓝 1
h' : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ W
v w s t :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa only [mul_assoc] using h _ (h' v v_in w w_in) _ (h' s s_in t t_in) | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u := by
rcases exists_nhds_one_split hu with ⟨W, W1, h⟩
rcases exists_nhds_one_split W1 with ⟨V, V1, h'⟩
use V, V1
intro v... | Mathlib.Topology.Algebra.Monoid.492_0.3p9EZf9ZWFxWOAq | @[to_additive exists_nhds_zero_quarter]
theorem exists_nhds_one_split4 {u : Set M} (hu : u ∈ 𝓝 (1 : M)) :
∃ V ∈ 𝓝 (1 : M), ∀ {v w s t}, v ∈ V → w ∈ V → s ∈ V → t ∈ V → v * w * s * t ∈ u | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
U : Set M
hU : U ∈ 𝓝 1
⊢ ∃ V, IsOpen V ∧ 1 ∈ V ∧ V * V ⊆ U | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases exists_open_nhds_one_split hU with ⟨V, Vo, V1, hV⟩ | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib.Topology.Algebra.Monoid.503_0.3p9EZf9ZWFxWOAq | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib_Topology_Algebra_Monoid |
case intro.intro.intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
U : Set M
hU : U ∈ 𝓝 1
V : Set M
Vo : IsOpen V
V1 : 1 ∈ V
hV : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
⊢ ∃ V, IsOpen V ∧ 1 ∈ V ∧ V * V ⊆ U | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | use V, Vo, V1 | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib.Topology.Algebra.Monoid.503_0.3p9EZf9ZWFxWOAq | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib_Topology_Algebra_Monoid |
case right
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
U : Set M
hU : U ∈ 𝓝 1
V : Set M
Vo : IsOpen V
V1 : 1 ∈ V
hV : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
⊢ V * V ⊆ U | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rintro _ ⟨x, y, hx, hy, rfl⟩ | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib.Topology.Algebra.Monoid.503_0.3p9EZf9ZWFxWOAq | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib_Topology_Algebra_Monoid |
case right.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
U : Set M
hU : U ∈ 𝓝 1
V : Set M
Vo : IsOpen V
V1 : 1 ∈ V
hV : ∀ v ∈ V, ∀ w ∈ V, v * w ∈ U
x y : M
hx : x ∈ V
hy : y ∈ V
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact hV _ hx _ hy | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib.Topology.Algebra.Monoid.503_0.3p9EZf9ZWFxWOAq | /-- Given a neighborhood `U` of `1` there is an open neighborhood `V` of `1`
such that `VV ⊆ U`. -/
@[to_additive "Given an open neighborhood `U` of `0` there is an open neighborhood `V` of `0`
such that `V + V ⊆ U`."]
theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
s t : Set M
hs : IsCompact s
ht : IsCompact t
⊢ IsCompact (s * t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [← image_mul_prod] | @[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) := by
| Mathlib.Topology.Algebra.Monoid.516_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
s t : Set M
hs : IsCompact s
ht : IsCompact t
⊢ IsCompact ((fun x => x.1 * x.2) '' s ×ˢ t) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact (hs.prod ht).image continuous_mul | @[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) := by
rw [← image_mul_prod]
| Mathlib.Topology.Algebra.Monoid.516_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem IsCompact.mul {s t : Set M} (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s * t) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → α → M
x : Filter α
a : ι → M
x✝ : ∀ i ∈ [], Tendsto (f i) x (𝓝 (a i))
⊢ Tendsto (fun b => List.prod (List.map (fun c => f c b) [])) x (𝓝 (List.prod... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp [tendsto_const_nhds] | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by | Mathlib.Topology.Algebra.Monoid.523_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
simp only [List.map_cons, Lis... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f✝ : ι → α → M
x : Filter α
a : ι → M
f : ι
l : List ι
h : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))
⊢ Tendsto (fun b => List.prod (List.map (fun c => f✝ c b... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [List.map_cons, List.prod_cons] | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
| Mathlib.Topology.Algebra.Monoid.523_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
simp only [List.map_cons, Lis... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f✝ : ι → α → M
x : Filter α
a : ι → M
f : ι
l : List ι
h : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))
⊢ Tendsto (fun b => f✝ f b * List.prod (List.map (fun c ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact
(h f (List.mem_cons_self _ _)).mul
(tendsto_list_prod l fun c hc => h c (List.mem_cons_of_mem _ hc)) | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
simp only [List.map_cons, Lis... | Mathlib.Topology.Algebra.Monoid.523_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem tendsto_list_prod {f : ι → α → M} {x : Filter α} {a : ι → M} :
∀ l : List ι,
(∀ i ∈ l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (l.map fun c => f c b).prod) x (𝓝 (l.map a).prod)
| [], _ => by simp [tendsto_const_nhds]
| f::l, h => by
simp only [List.map_cons, Lis... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
⊢ ContinuousOn (fun a => List.prod (List.map (fun i => f i a) l)) t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | intro x hx | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
| Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
x : X
hx : x ∈ t
⊢ ContinuousWithinAt (fun a => List.prod (List.map (fun i => f i a) l))... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [continuousWithinAt_iff_continuousAt_restrict _ hx] | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
| Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
x : X
hx : x ∈ t
⊢ ContinuousAt (restrict t fun a => List.prod (List.map (fun i => f i a... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine' tendsto_list_prod _ fun i hi => _ | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
| Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
h : ∀ i ∈ l, ContinuousOn (f i) t
x : X
hx : x ∈ t
i : ι
hi : i ∈ l
⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | specialize h i hi x hx | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
refine' tendsto_list_prod _ fun i hi => _
| Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
x : X
hx : x ∈ t
i : ι
hi : i ∈ l
h : ContinuousWithinAt (f i) t x
⊢ Tendsto (fun b => f i ↑b) (𝓝 { val := x, property :=... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [continuousWithinAt_iff_continuousAt_restrict _ hx] at h | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
refine' tendsto_list_prod _ fun i hi => _
specialize ... | Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
f : ι → X → M
l : List ι
t : Set X
x : X
hx : x ∈ t
i : ι
hi : i ∈ l
h : ContinuousAt (restrict t (f i)) { val := x, property := hx }
⊢ Tendsto (fun b => f i... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact h | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t := by
intro x hx
rw [continuousWithinAt_iff_continuousAt_restrict _ hx]
refine' tendsto_list_prod _ fun i hi => _
specialize ... | Mathlib.Topology.Algebra.Monoid.545_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_list_prod {f : ι → X → M} (l : List ι) {t : Set X}
(h : ∀ i ∈ l, ContinuousOn (f i) t) :
ContinuousOn (fun a => (l.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
⊢ Continuous fun a => a ^ 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using continuous_const | @[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
| 0 => by | Mathlib.Topology.Algebra.Monoid.558_0.3p9EZf9ZWFxWOAq | @[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous fun a => a ^ (k + 1) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [pow_succ] | @[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
| 0 => by simpa using continuous_const
| k + 1 => by
| Mathlib.Topology.Algebra.Monoid.558_0.3p9EZf9ZWFxWOAq | @[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
k : ℕ
⊢ Continuous fun a => a * a ^ k | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuous_id.mul (continuous_pow _) | @[to_additive (attr := continuity)]
theorem continuous_pow : ∀ n : ℕ, Continuous fun a : M => a ^ n
| 0 => by simpa using continuous_const
| k + 1 => by
simp only [pow_succ]
| Mathlib.Topology.Algebra.Monoid.558_0.3p9EZf9ZWFxWOAq | @[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : b * a = 1
⊢ Tendsto (fun x => a * x) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_left b)) | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) := by
| Mathlib.Topology.Algebra.Monoid.631_0.3p9EZf9ZWFxWOAq | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : b * a = 1
⊢ Tendsto ((fun b_1 => b * b_1) ∘ fun x => a * x) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [comp_mul_left, ha, one_mul] | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.Tend... | Mathlib.Topology.Algebra.Monoid.631_0.3p9EZf9ZWFxWOAq | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : b * a = 1
⊢ Tendsto (fun x => x) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact Filter.tendsto_id | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.Tend... | Mathlib.Topology.Algebra.Monoid.631_0.3p9EZf9ZWFxWOAq | /-- Left-multiplication by a left-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : a * b = 1
⊢ Tendsto (fun x => x * a) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_right b)) | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) := by
| Mathlib.Topology.Algebra.Monoid.645_0.3p9EZf9ZWFxWOAq | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : a * b = 1
⊢ Tendsto ((fun b_1 => b_1 * b) ∘ fun x => x * a) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [comp_mul_right, ha, mul_one] | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.T... | Mathlib.Topology.Algebra.Monoid.645_0.3p9EZf9ZWFxWOAq | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : Monoid M
inst✝ : ContinuousMul M
a b : M
ha : a * b = 1
⊢ Tendsto (fun x => x) (cocompact M) (cocompact M) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact Filter.tendsto_id | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) := by
refine Filter.T... | Mathlib.Topology.Algebra.Monoid.645_0.3p9EZf9ZWFxWOAq | /-- Right-multiplication by a right-invertible element of a topological monoid is proper, i.e.,
inverse images of compact sets are compact. -/
theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : IsScalarTower R A A
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousMul A
q : R
⊢ Conti... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp (config := { singlePass := true }) only [← smul_one_mul q (_ : A)] | /-- If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[Algebra R A]` is available. -/
@[to_additive "If `R` acts on `A` via `A`, then continuous addition implies
continuous affine addition by constant... | Mathlib.Topology.Algebra.Monoid.655_0.3p9EZf9ZWFxWOAq | /-- If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[Algebra R A]` is available. -/
@[to_additive "If `R` acts on `A` via `A`, then continuous addition implies
continuous affine addition by constant... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : IsScalarTower R A A
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousMul A
q : R
⊢ Conti... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuous_const.mul continuous_id | /-- If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[Algebra R A]` is available. -/
@[to_additive "If `R` acts on `A` via `A`, then continuous addition implies
continuous affine addition by constant... | Mathlib.Topology.Algebra.Monoid.655_0.3p9EZf9ZWFxWOAq | /-- If `R` acts on `A` via `A`, then continuous multiplication implies continuous scalar
multiplication by constants.
Notably, this instances applies when `R = A`, or when `[Algebra R A]` is available. -/
@[to_additive "If `R` acts on `A` via `A`, then continuous addition implies
continuous affine addition by constant... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : SMulCommClass R A A
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousMul A
q : R
⊢ Conti... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp (config := { singlePass := true }) only [← mul_smul_one q (_ : A)] | /-- If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ`.-/
@[to_additive "If the action of `R` on `A` commutes with left-addition, then
continuous addition implies continuous... | Mathlib.Topology.Algebra.Monoid.669_0.3p9EZf9ZWFxWOAq | /-- If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ`.-/
@[to_additive "If the action of `R` on `A` commutes with left-addition, then
continuous addition implies continuous... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Monoid M
inst✝⁵ : ContinuousMul M
R : Type u_6
A : Type u_7
inst✝⁴ : Monoid A
inst✝³ : SMul R A
inst✝² : SMulCommClass R A A
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousMul A
q : R
⊢ Conti... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuous_id.mul continuous_const | /-- If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ`.-/
@[to_additive "If the action of `R` on `A` commutes with left-addition, then
continuous addition implies continuous... | Mathlib.Topology.Algebra.Monoid.669_0.3p9EZf9ZWFxWOAq | /-- If the action of `R` on `A` commutes with left-multiplication, then continuous multiplication
implies continuous scalar multiplication by constants.
Notably, this instances applies when `R = Aᵐᵒᵖ`.-/
@[to_additive "If the action of `R` on `A` commutes with left-addition, then
continuous addition implies continuous... | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → α → M
x : Filter α
a : ι → M
s : Multiset ι
⊢ (∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => Multiset.prod (Multiset.map (fun c =>... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases s with ⟨l⟩ | @[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) := by
| Mathlib.Topology.Algebra.Monoid.737_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → α → M
x : Filter α
a : ι → M
s : Multiset ι
l : List ι
⊢ (∀ i ∈ Quot.mk Setoid.r l, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => M... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using tendsto_list_prod l | @[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) := by
rcases s with ⟨l⟩
| Mathlib.Topology.Algebra.Monoid.737_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
⊢ (∀ i ∈ s, Continuous (f i)) → Continuous fun a => Multiset.prod (Multiset.map (fun i => f i a) s) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases s with ⟨l⟩ | @[to_additive (attr := continuity)]
theorem continuous_multiset_prod {f : ι → X → M} (s : Multiset ι) :
(∀ i ∈ s, Continuous (f i)) → Continuous fun a => (s.map fun i => f i a).prod := by
| Mathlib.Topology.Algebra.Monoid.754_0.3p9EZf9ZWFxWOAq | @[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
l : List ι
⊢ (∀ i ∈ Quot.mk Setoid.r l, Continuous (f i)) →
Continuous fun a => Multiset.prod (Multiset.map (fun... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using continuous_list_prod l | @[to_additive (attr := continuity)]
theorem continuous_multiset_prod {f : ι → X → M} (s : Multiset ι) :
(∀ i ∈ s, Continuous (f i)) → Continuous fun a => (s.map fun i => f i a).prod := by
rcases s with ⟨l⟩
| Mathlib.Topology.Algebra.Monoid.754_0.3p9EZf9ZWFxWOAq | @[to_additive (attr | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
t : Set X
⊢ (∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => Multiset.prod (Multiset.map (fun i => f i a) s)) t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases s with ⟨l⟩ | @[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t := by
| Mathlib.Topology.Algebra.Monoid.762_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
case mk
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
s : Multiset ι
t : Set X
l : List ι
⊢ (∀ i ∈ Quot.mk Setoid.r l, ContinuousOn (f i) t) →
ContinuousOn (fun a => Multiset.prod (... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simpa using continuousOn_list_prod l | @[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t := by
rcases s with ⟨l⟩
| Mathlib.Topology.Algebra.Monoid.762_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X✝ : Type u_5
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ i ∈ s, f i =ᶠ[l] g i
⊢ ∏ i in s, f i =ᶠ[l] ∏ i in ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i := by
| Mathlib.Topology.Algebra.Monoid.784_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i | Mathlib_Topology_Algebra_Monoid |
case hs
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X✝ : Type u_5
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ i ∈ s, f i =ᶠ[l] g i
⊢ ∀ᶠ (x : X) in l, ∀ ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rwa [eventually_all_finset] | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i := by
replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x
· | Mathlib.Topology.Algebra.Monoid.784_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X✝ : Type u_5
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ᶠ (x : X) in l, ∀ i ∈ s, f i x = g i x
⊢ ∏ i in s, ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | filter_upwards [hs] with x hx | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i := by
replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x
· rwa [eventually_all_finset]
| Mathlib.Topology.Algebra.Monoid.784_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i | Mathlib_Topology_Algebra_Monoid |
case h
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X✝ : Type u_5
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ᶠ (x : X) in l, ∀ i ∈ s, f i x = g i x
x : X... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [Finset.prod_apply, Finset.prod_congr rfl hx] | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i := by
replace hs : ∀ᶠ x in l, ∀ i ∈ s, f i x = g i x
· rwa [eventually_all_finset]
filter_upwards [hs] with x hx
| Mathlib.Topology.Algebra.Monoid.784_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i in s, f i =ᶠ[l] ∏ i in s, g i | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i =>... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases hf x₀ with ⟨U, hxU, hUf⟩ | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I := by
| Mathlib.Topology.Algebra.Monoid.796_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I | Mathlib_Topology_Algebra_Monoid |
case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine' ⟨hUf.toFinset, mem_of_superset hxU fun y hy i hi => _⟩ | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I := by
rcases hf x₀ with ⟨U, hxU, hUf⟩
| Mathlib.Topology.Algebra.Monoid.796_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I | Mathlib_Topology_Algebra_Monoid |
case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [hUf.coe_toFinset] | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I := by
rcases hf x₀ with ⟨U, hxU, hUf⟩
refine' ⟨hUf.toFinset, mem_of_superset hxU ... | Mathlib.Topology.Algebra.Monoid.796_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I | Mathlib_Topology_Algebra_Monoid |
case intro.intro
ι : Type u_1
α : Type u_2
M✝ : Type u_3
N : Type u_4
X : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace M✝
inst✝² : CommMonoid M✝
inst✝¹ : ContinuousMul M✝
M : Type u_6
inst✝ : CommMonoid M
f : ι → X → M
hf : LocallyFinite fun i => mulSupport (f i)
x₀ : X
U : Set X
hxU : U ∈ 𝓝 x₀
hUf :... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact ⟨y, hi, hy⟩ | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I := by
rcases hf x₀ with ⟨U, hxU, hUf⟩
refine' ⟨hUf.toFinset, mem_of_superset hxU ... | Mathlib.Topology.Algebra.Monoid.796_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem LocallyFinite.exists_finset_mulSupport {M : Type*} [CommMonoid M] {f : ι → X → M}
(hf : LocallyFinite fun i => mulSupport <| f i) (x₀ : X) :
∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, (mulSupport fun i => f i x) ⊆ I | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
hc : ∀ (i : ι), Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ Continuous fun x => ∏ᶠ (i : ι), f i x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine' continuous_iff_continuousAt.2 fun x => _ | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x := by
| Mathlib.Topology.Algebra.Monoid.816_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
hc : ∀ (i : ι), Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
x : X
⊢ ContinuousAt (fun x => ∏ᶠ (i : ι), f i x) x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rcases finprod_eventually_eq_prod hf x with ⟨s, hs⟩ | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x := by
refine' continuous_iff_continuousAt.2 fun x => _
| Mathlib.Topology.Algebra.Monoid.816_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x | Mathlib_Topology_Algebra_Monoid |
case intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
hc : ∀ (i : ι), Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
x : X
s : Finset ι
hs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine' ContinuousAt.congr _ (EventuallyEq.symm hs) | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x := by
refine' continuous_iff_continuousAt.2 fun x => _
rcases finprod_eventually_eq_prod hf x with ⟨s, hs⟩
| Mathlib.Topology.Algebra.Monoid.816_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x | Mathlib_Topology_Algebra_Monoid |
case intro
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
hc : ∀ (i : ι), Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
x : X
s : Finset ι
hs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact tendsto_finset_prod _ fun i _ => (hc i).continuousAt | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x := by
refine' continuous_iff_continuousAt.2 fun x => _
rcases finprod_eventually_eq_prod hf x with ⟨s, hs⟩
refine' ContinuousAt.congr _ (Event... | Mathlib.Topology.Algebra.Monoid.816_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
p : ι → Prop
hc : ∀ (i : ι), p i → Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ Continuous fun x => ∏ᶠ (i : ι) (_ : p i)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | simp only [← finprod_subtype_eq_finprod_cond] | @[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x := by
| Mathlib.Topology.Algebra.Monoid.826_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace M
inst✝¹ : CommMonoid M
inst✝ : ContinuousMul M
f : ι → X → M
p : ι → Prop
hc : ∀ (i : ι), p i → Continuous (f i)
hf : LocallyFinite fun i => mulSupport (f i)
⊢ Continuous fun x => ∏ᶠ (j : { i // p i }... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective) | @[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x := by
simp only [← finprod_subtype_eq_finprod_cond]
| Mathlib.Topology.Algebra.Monoid.826_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
(hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
ts : ι' → TopologicalSpace M
h' : ∀ (i : ι'), ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [← sInf_range] | @[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ := by
| Mathlib.Topology.Algebra.Monoid.857_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
ts : ι' → TopologicalSpace M
h' : ∀ (i : ι'), ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | exact continuousMul_sInf (Set.forall_range_iff.mpr h') | @[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ := by
rw [← sInf_range]
| Mathlib.Topology.Algebra.Monoid.857_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | rw [inf_eq_iInf] | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
| Mathlib.Topology.Algebra.Monoid.865_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | refine' continuousMul_iInf fun b => _ | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
| Mathlib.Topology.Algebra.Monoid.865_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ | Mathlib_Topology_Algebra_Monoid |
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
b : Bool
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | cases b | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine' continuousMul_iInf fun b => _
| Mathlib.Topology.Algebra.Monoid.865_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ | Mathlib_Topology_Algebra_Monoid |
case false
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | assumption | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine' continuousMul_iInf fun b => _
cases b <;> | Mathlib.Topology.Algebra.Monoid.865_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ | Mathlib_Topology_Algebra_Monoid |
case true
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝¹ : TopologicalSpace X
ι' : Sort u_6
inst✝ : Mul M
t₁ t₂ : TopologicalSpace M
h₁ : ContinuousMul M
h₂ : ContinuousMul M
⊢ ContinuousMul M | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | assumption | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine' continuousMul_iInf fun b => _
cases b <;> | Mathlib.Topology.Algebra.Monoid.865_0.3p9EZf9ZWFxWOAq | @[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ | Mathlib_Topology_Algebra_Monoid |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
⊢ (fun i => IsLimit.lift (hs... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | apply IsLimit.hom_ext (hs i') | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
⊢ ∀ (j : J),
((fun i => ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | intro j | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
j : J
⊢ ((fun i => IsLimit.l... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | have eq := fun k => (hs k).fac ((eval C c k).mapCone t) | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
j : J
eq :
∀ (k : ι) (j : ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | simp only [Functor.mapCone_π_app, eval_map] at eq | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
j : J
eq : ∀ (k : ι) (j : J)... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | simp only [Functor.mapCone_π_app, eval_map, assoc] | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
i i' : ι
x✝ : ComplexShape.Rel c i i'
j : J
eq : ∀ (k : ι) (j : J)... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | rw [eq i', ← Hom.comm, reassoc_of% (eq i), Hom.comm] | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
j : J
⊢ (fun t => Hom.mk fun i => IsLimit.lift (hs i) ((eval C c i... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | ext i | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case h
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
j : J
i : ι
⊢ Hom.f ((fun t => Hom.mk fun i => IsLimit.lift... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | apply (hs i).fac | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
m : t.pt ⟶ s.pt
hm : ∀ (j : J), m ≫ s.π.app j = t.π.app j
⊢ m = (f... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | ext i | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case h
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
m : t.pt ⟶ s.pt
hm : ∀ (j : J), m ≫ s.π.app j = t.π.app j
i... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | apply (hs i).uniq ((eval C c i).mapCone t) | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case h.x
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
m : t.pt ⟶ s.pt
hm : ∀ (j : J), m ≫ s.π.app j = t.π.app j... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | intro j | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case h.x
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
m : t.pt ⟶ s.pt
hm : ∀ (j : J), m ≫ s.π.app j = t.π.app j... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | dsimp | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case h.x
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝² : Category.{?u.152, u_1} C
inst✝¹ : Category.{?u.156, u_3} J
c : ComplexShape ι
inst✝ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
s : Cone F
hs : (i : ι) → IsLimit ((eval C c i).mapCone s)
t : Cone F
m : t.pt ⟶ s.pt
hm : ∀ (j : J), m ≫ s.π.app j = t.π.app j... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | simp only [← comp_f, hm] | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t :=
{ f := fun i => (hs i).lift ((eval C c i).... | Mathlib.Algebra.Homology.HomologicalComplexLimits.30_0.gJN7GlsIU4rmUTz | /-- A cone in `HomologicalComplex C c` is limit if the induced cones obtained
by applying `eval C c i : HomologicalComplex C c ⥤ C` for all `i` are limit. -/
def isLimitOfEval (s : Cone F)
(hs : ∀ (i : ι), IsLimit ((eval C c i).mapCone s)) : IsLimit s where
lift t | Mathlib_Algebra_Homology_HomologicalComplexLimits |
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝³ : Category.{?u.5993, u_1} C
inst✝² : Category.{?u.5997, u_3} J
c : ComplexShape ι
inst✝¹ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝ : ∀ (n : ι), HasLimit (F ⋙ eval C c n)
n m : ι
h : ¬ComplexShape.Rel c n m
⊢ (fun n m => limMap (NatTrans.mk fun j => d (F.obj ... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | ext j | /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F where
pt :=
{ X := fun n => limit (F ⋙ eval C c n)
d := fun n m => limMap { app := fun j => (F.obj j).d n m }
shape := fun {n ... | Mathlib.Algebra.Homology.HomologicalComplexLimits.55_0.gJN7GlsIU4rmUTz | /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F where
pt | Mathlib_Algebra_Homology_HomologicalComplexLimits |
case w
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝³ : Category.{?u.5993, u_1} C
inst✝² : Category.{?u.5997, u_3} J
c : ComplexShape ι
inst✝¹ : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝ : ∀ (n : ι), HasLimit (F ⋙ eval C c n)
n m : ι
h : ¬ComplexShape.Rel c n m
j : J
⊢ (fun n m => limMap (NatTrans.mk fun j... | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
/-!
# Limits ... | rw [limMap_π] | /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F where
pt :=
{ X := fun n => limit (F ⋙ eval C c n)
d := fun n m => limMap { app := fun j => (F.obj j).d n m }
shape := fun {n ... | Mathlib.Algebra.Homology.HomologicalComplexLimits.55_0.gJN7GlsIU4rmUTz | /-- A cone for a functor `F : J ⥤ HomologicalComplex C c` which is given in degree `n` by
the limit `F ⋙ eval C c n`. -/
@[simps]
noncomputable def coneOfHasLimitEval : Cone F where
pt | Mathlib_Algebra_Homology_HomologicalComplexLimits |
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