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 |
|---|---|---|---|---|---|---|
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | left | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
· | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inl.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
· left; | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | right | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
· left; rfl
· | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inr.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact exists_apply_eq_finset_sup p hs x | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
· left; rfl
· right; | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext x | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [smul_apply, finset_sup_apply, finset_sup_apply] | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by
ext x
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | symm | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.mul_finset_sup C s (fun i ↦ ⟨p i x, map_nonneg _ _⟩)) | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i := by
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' Finset.sup_le_iff.mpr _ | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i := by
classical
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro i hi | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left] | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact bot_le | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | lift a to ℝ≥0 using ha | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a := by
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [finset_sup_apply, NNReal.coe_le_coe] | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a := by
lift a to ℝ≥0 using ha
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact Finset.sup_le h | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a := by
lift a to ℝ≥0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 ≤ a)
(h : ∀ i, i ∈ s → p i x ≤ a) : s.sup p x ≤ a | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | lift a to ℝ≥0 using ha.le | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
| Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff] | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
lift a to ℝ≥0 using ha.le
| Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact h | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
lift a to ℝ≥0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
· | Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : �... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact NNReal.coe_pos.mpr ha | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
lift a to ℝ≥0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
· exact h
· | Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : SeminormedCommRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply] | theorem comp_smul (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) :
p.comp (c • f) = ‖c‖₊ • p.comp f :=
ext fun _ => by
| Mathlib.Analysis.Seminorm.464_0.ywwMCgoKeIFKDZ3 | theorem comp_smul (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (c : 𝕜₂) :
p.comp (c • f) = ‖c‖₊ • p.comp f | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
⊢ 0 ∈ lowerBounds (range fun u => p u + q (x - u)) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rintro _ ⟨x, rfl⟩ | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
⟨0, by
| Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ x : E
⊢ 0 ≤ (fun u => p u + q (x✝ - u)) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | dsimp | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
⟨0, by
rintro _ ⟨x, rfl⟩
| Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ x : E
⊢ 0 ≤ p x + q (x✝ - x) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | positivity | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
⟨0, by
rintro _ ⟨x, rfl⟩
dsimp; | Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E := p.toAdd... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro a x | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E := p.toAd... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | obtain rfl | ha := eq_or_ne a 0 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [norm_zero, zero_mul, zero_smul] | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => ⟨0, by rwa [map_zero, sub_zero, map_zero, add_zero]⟩ | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝¹ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E := p.toA... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rwa [map_zero, sub_zero, map_zero, add_zero] | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
refine'
... | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, ← map_smul_eq_mul p, ←
map_smul_eq_mul q, smul_sub] | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
refine'
... | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine'
Function.Surjective.iInf_congr ((a⁻¹ • ·) : E → E)
(fun u => ⟨a • u, inv_smul_smul₀ ha u⟩) fun u => _ | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
refine'
... | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [smul_inv_smul₀ ha] | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q :=
{ p.toAddGroupSeminorm ⊓ q.toAddGroupSeminorm with
toFun := fun x => ⨅ u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
· rw [norm_zero, zero_mul, zero_smul]
refine'
... | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm 𝕜 E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
src✝ : SemilatticeSup (Seminorm 𝕜 E) := instSemilattic... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [sub_self, map_zero, add_zero] | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) :=
{ Seminorm.instSemilatticeSup with
inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
| Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
src✝ : SemilatticeSup (Seminorm 𝕜 E) := instSemilattic... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) :=
{ Seminorm.instSemilatticeSup with
inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; | Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
src✝ : SemilatticeSup (Seminorm 𝕜 E) := instSemilattic... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [sub_self, map_zero, zero_add, sub_zero] | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) :=
{ Seminorm.instSemilatticeSup with
inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_ran... | Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
src✝ : SemilatticeSup (Seminorm 𝕜 E) := instSemilattic... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) :=
{ Seminorm.instSemilatticeSup with
inf := (· ⊓ ·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_ran... | Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x : E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : IsScalarT... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem smul_inf [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q := by
| Mathlib.Analysis.Seminorm.526_0.ywwMCgoKeIFKDZ3 | theorem smul_inf [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x : E
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ : Is... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [smul_apply, inf_apply, smul_apply, ← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add] | theorem smul_inf [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q := by
ext
| Mathlib.Analysis.Seminorm.526_0.ywwMCgoKeIFKDZ3 | theorem smul_inf [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊓ q) = r • p ⊓ r • q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
⊢ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [iSup_apply, ← @Real.ciSup_const_zero s] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
⊢ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | congr! | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLik... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLik... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact map_zero i.1 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
x... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases h with ⟨q, hq⟩ | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ
hq ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | obtain rfl | h := s.eq_empty_or_nonempty | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inl
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ x y : E
q : E → ℝ
hq : q ∈ upperBounds (FunLi... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [Real.ciSup_empty] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | haveI : Nonempty ↑s := h.coe_sort | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E
q : E → ℝ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' ciSup_le fun i =>
((i : Seminorm 𝕜 E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype... | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_upperBounds, forall_range_iff] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_upperBounds, forall_range_iff] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact fun j => hq (mem_image_of_mem _ j.2) _ | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
x y : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact fun j => hq (mem_image_of_mem _ j.2) _ | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
x... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
x... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | congr! 2 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝¹ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunL... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLi... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact i.1.neg' _ | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
a... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
a... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm 𝕜 E) x] | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLike.coe '' s)
a... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | congr! | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝¹ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunL... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
h : BddAbove (FunLi... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact i.1.smul' a x | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm 𝕜 E` as follows:
* if `s` is `BddAbove` *as a set of functions `E → ℝ`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
H : BddAbove (FunLike.coe '' s)
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | dsimp | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) :=
⟨fun ⟨q, hq⟩ => ⟨q, ball_image_of_ball fun p hp => hq hp⟩, fun H =>
⟨sSup s, fun p hp x => by
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
H : BddAbove (FunLike.coe '' s)
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.coe_sSup_eq' H, iSup_apply] | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) :=
⟨fun ⟨q, hq⟩ => ⟨q, ball_image_of_ball fun p hp => hq hp⟩, fun H =>
⟨sSup s, fun p hp x => by
dsimp
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
H : BddAbove (FunLike.coe '' s)
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases H with ⟨q, hq⟩ | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) :=
⟨fun ⟨q, hq⟩ => ⟨q, ball_image_of_ball fun p hp => hq hp⟩, fun H =>
⟨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
p : Seminorm 𝕜 E
hp... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact
le_ciSup ⟨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) x⟩ ⟨p, hp⟩ | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) :=
⟨fun ⟨q, hq⟩ => ⟨q, ball_image_of_ball fun p hp => hq hp⟩, fun H =>
⟨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with ⟨q, hq⟩
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm 𝕜 E} :
BddAbove s ↔ BddAbove ((↑) '' s : Set (E → ℝ)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
p : ι → Seminorm 𝕜 E
⊢ BddAbove (range p) ↔ ∀ (x : E), B... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.bddAbove_iff, ← range_comp, bddAbove_range_pi] | protected theorem bddAbove_range_iff {p : ι → Seminorm 𝕜 E} :
BddAbove (range p) ↔ ∀ x, BddAbove (range fun i ↦ p i x) := by
| Mathlib.Analysis.Seminorm.606_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_range_iff {p : ι → Seminorm 𝕜 E} :
BddAbove (range p) ↔ ∀ x, BddAbove (range fun i ↦ p i x) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
p : ι → Seminorm 𝕜 E
⊢ (∀ (a : E), BddAbove (range fun i... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | protected theorem bddAbove_range_iff {p : ι → Seminorm 𝕜 E} :
BddAbove (range p) ↔ ∀ x, BddAbove (range fun i ↦ p i x) := by
rw [Seminorm.bddAbove_iff, ← range_comp, bddAbove_range_pi]; | Mathlib.Analysis.Seminorm.606_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_range_iff {p : ι → Seminorm 𝕜 E} :
BddAbove (range p) ↔ ∀ x, BddAbove (range fun i ↦ p i x) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
ι : Type u_13
p : ι → Seminorm 𝕜 E
hp : BddAbove (range... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← sSup_range, Seminorm.coe_sSup_eq hp] | protected theorem coe_iSup_eq {ι : Type*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) := by
| Mathlib.Analysis.Seminorm.615_0.ywwMCgoKeIFKDZ3 | protected theorem coe_iSup_eq {ι : Type*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
ι : Type u_13
p : ι → Seminorm 𝕜 E
hp : BddAbove (range... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact iSup_range' (fun p : Seminorm 𝕜 E => (p : E → ℝ)) p | protected theorem coe_iSup_eq {ι : Type*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) := by
rw [← sSup_range, Seminorm.coe_sSup_eq hp]
| Mathlib.Analysis.Seminorm.615_0.ywwMCgoKeIFKDZ3 | protected theorem coe_iSup_eq {ι : Type*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) :
↑(⨆ i, p i) = ⨆ i, ((p i : Seminorm 𝕜 E) : E → ℝ) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hp : BddAbove s
x : E
⊢ (sSup s) ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.coe_sSup_eq hp, iSup_apply] | protected theorem sSup_apply {s : Set (Seminorm 𝕜 E)} (hp : BddAbove s) {x : E} :
(sSup s) x = ⨆ p : s, (p : E → ℝ) x := by
| Mathlib.Analysis.Seminorm.621_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_apply {s : Set (Seminorm 𝕜 E)} (hp : BddAbove s) {x : E} :
(sSup s) x = ⨆ p : s, (p : E → ℝ) x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
ι : Type u_13
p : ι → Seminorm 𝕜 E
hp : BddAbove (rang... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.coe_iSup_eq hp, iSup_apply] | protected theorem iSup_apply {ι : Type*} {p : ι → Seminorm 𝕜 E}
(hp : BddAbove (range p)) {x : E} : (⨆ i, p i) x = ⨆ i, p i x := by
| Mathlib.Analysis.Seminorm.625_0.ywwMCgoKeIFKDZ3 | protected theorem iSup_apply {ι : Type*} {p : ι → Seminorm 𝕜 E}
(hp : BddAbove (range p)) {x : E} : (⨆ i, p i) x = ⨆ i, p i x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
⊢ sSup ∅ = ⊥ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ := by
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x x✝ : E
⊢ (sSup ∅) x✝ = ⊥ x✝ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty] | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ := by
ext
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x x✝ : E
⊢ 0 = ⊥ x✝ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (∅ : Set (Seminorm 𝕜 E)) = ⊥ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q : Seminorm 𝕜 E
x : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
hs₂ : Set.Nonempt... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' ⟨fun p hp x => _, fun p hp x => _⟩ | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | haveI : Nonempty ↑s := hs₂.coe_sort | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | haveI : Nonempty ↑s := hs₂.coe_sort | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | dsimp | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | dsimp | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.coe_sSup_eq hs₁, iSup_apply] | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Seminorm.coe_sSup_eq hs₁, iSup_apply] | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases hs₁ with ⟨q, hq⟩ | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hs₁, iSup_apply]
· | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q✝ : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₂ : Set.... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact le_ciSup ⟨q x, forall_range_iff.mpr fun i : s => hq i.2 x⟩ ⟨p, hp⟩ | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hs₁, iSup_apply]
· rcases hs₁ with ⟨q, hq⟩
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x✝ : E
s : Set (Seminorm 𝕜 E)
hs₁ : BddAbove s
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact ciSup_le fun q => hp q.2 x | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' ⟨fun p hp x => _, fun p hp x => _⟩ <;> haveI : Nonempty ↑s := hs₂.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hs₁, iSup_apply]
· rcases hs₁ with ⟨q, hq⟩
exact le_ciSup... | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm 𝕜 E)) (hs₁ : BddAbove s) (hs₂ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x y : E
r : ℝ
hr : 0 < r
⊢ x ∈ ball p x r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [hr] | theorem mem_ball_self (hr : 0 < r) : x ∈ ball p x r := by | Mathlib.Analysis.Seminorm.696_0.ywwMCgoKeIFKDZ3 | theorem mem_ball_self (hr : 0 < r) : x ∈ ball p x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x y : E
r : ℝ
hr : 0 ≤ r
⊢ x ∈ closedBall p x r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [hr] | theorem mem_closedBall_self (hr : 0 ≤ r) : x ∈ closedBall p x r := by | Mathlib.Analysis.Seminorm.699_0.ywwMCgoKeIFKDZ3 | theorem mem_closedBall_self (hr : 0 ≤ r) : x ∈ closedBall p x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x y : E
r : ℝ
⊢ y ∈ ball p 0 r ↔ p y < r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_ball, sub_zero] | theorem mem_ball_zero : y ∈ ball p 0 r ↔ p y < r := by | Mathlib.Analysis.Seminorm.702_0.ywwMCgoKeIFKDZ3 | theorem mem_ball_zero : y ∈ ball p 0 r ↔ p y < r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x y : E
r : ℝ
⊢ y ∈ closedBall p 0 r ↔ p y ≤ r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_closedBall, sub_zero] | theorem mem_closedBall_zero : y ∈ closedBall p 0 r ↔ p y ≤ r := by | Mathlib.Analysis.Seminorm.705_0.ywwMCgoKeIFKDZ3 | theorem mem_closedBall_zero : y ∈ closedBall p 0 r ↔ p y ≤ r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x✝ y : E
r✝ : ℝ
x : E
r : ℝ
⊢ closedBall p x r = ⋂ ρ, ⋂ (_ : ρ > ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext y | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = ⋂ ρ > r, ball p x ρ := by
| Mathlib.Analysis.Seminorm.720_0.ywwMCgoKeIFKDZ3 | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = ⋂ ρ > r, ball p x ρ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x✝ y✝ : E
r✝ : ℝ
x : E
r : ℝ
y : E
⊢ y ∈ closedBall p x r ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [mem_closedBall, mem_iInter₂, mem_ball, ← forall_lt_iff_le'] | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = ⋂ ρ > r, ball p x ρ := by
ext y; | Mathlib.Analysis.Seminorm.720_0.ywwMCgoKeIFKDZ3 | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = ⋂ ρ > r, ball p x ρ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x✝ y : E
r : ℝ
x : E
hr : 0 < r
⊢ ball 0 x r = univ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Set.eq_univ_iff_forall, ball] | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm 𝕜 E) x r = Set.univ := by
| Mathlib.Analysis.Seminorm.724_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm 𝕜 E) x r = Set.univ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
x✝ y : E
r : ℝ
x : E
hr : 0 < r
⊢ ∀ (x_1 : E), x_1 ∈ {y | 0 (y - ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [hr] | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm 𝕜 E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
| Mathlib.Analysis.Seminorm.724_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm 𝕜 E) x r = Set.univ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x✝ y : E
r✝ : ℝ
p : Seminorm 𝕜 E
c : ℝ≥0
hc : 0 < c
r : ℝ
x : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) := by
| Mathlib.Analysis.Seminorm.735_0.ywwMCgoKeIFKDZ3 | theorem ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x✝¹ y : E
r✝ : ℝ
p : Seminorm 𝕜 E
c : ℝ≥0
hc : 0 < c
r :... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)] | theorem ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) := by
ext
| Mathlib.Analysis.Seminorm.735_0.ywwMCgoKeIFKDZ3 | theorem ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).ball x r = p.ball x (r / c) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x✝ y : E
r✝ : ℝ
p : Seminorm 𝕜 E
c : ℝ≥0
hc : 0 < c
r : ℝ
x : E... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | ext | theorem closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closedBall x r = p.closedBall x (r / c) := by
| Mathlib.Analysis.Seminorm.742_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closedBall x r = p.closedBall x (r / c) | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x✝¹ y : E
r✝ : ℝ
p : Seminorm 𝕜 E
c : ℝ≥0
hc : 0 < c
r :... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)] | theorem closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closedBall x r = p.closedBall x (r / c) := by
ext
| Mathlib.Analysis.Seminorm.742_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
(c • p).closedBall x r = p.closedBall x (r / c) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p q : Seminorm 𝕜 E
e : E
r : ℝ
⊢ ball (p ⊔ q) e ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [ball, ← Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff] | theorem ball_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
ball (p ⊔ q) e r = ball p e r ∩ ball q e r := by
| Mathlib.Analysis.Seminorm.749_0.ywwMCgoKeIFKDZ3 | theorem ball_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
ball (p ⊔ q) e r = ball p e r ∩ ball q e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p q : Seminorm 𝕜 E
e : E
r : ℝ
⊢ closedBall (p ⊔... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [closedBall, ← Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff] | theorem closedBall_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
closedBall (p ⊔ q) e r = closedBall p e r ∩ closedBall q e r := by
| Mathlib.Analysis.Seminorm.754_0.ywwMCgoKeIFKDZ3 | theorem closedBall_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
closedBall (p ⊔ q) e r = closedBall p e r ∩ closedBall q e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
H : Finset.Non... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
| Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
case h₀
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
x y : E
r✝ : ℝ
p : ι → Seminorm 𝕜 E
s : Finset ι
e : E
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | classical simp | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
· | Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ι → Seminorm 𝕜 E) (s : Finset ι) (H : s.Nonempty) (e : E) (r : ℝ) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
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