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 |
|---|---|---|---|---|---|---|
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
⊢ mk ⁻¹' (mk '' s) ⊆ s | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rintro x ⟨y, hys, hxy⟩ | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
| Mathlib.Topology.Inseparable.474_0.2NeLzt0mQ64QlfB | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s | Mathlib_Topology_Inseparable |
case intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
x y : X
hys : y... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
refine' Subset.antisymm _ (subset_preimage_image _ _)
rintro x ⟨y, hys, hxy⟩
| Mathlib.Topology.Inseparable.474_0.2NeLzt0mQ64QlfB | theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ IsClosed (Set.range mk) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [range_mk] | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by | Mathlib.Topology.Inseparable.485_0.2NeLzt0mQ64QlfB | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ IsClosed univ | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | exact isClosed_univ | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
inducing_mk.isClosedMap <| by rw [range_mk]; | Mathlib.Topology.Inseparable.485_0.2NeLzt0mQ64QlfB | theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ Filter.map mk (𝓝 x) = 𝓝 (mk x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk] | theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) := by
| Mathlib.Topology.Inseparable.499_0.2NeLzt0mQ64QlfB | theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ Filter.map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk] | theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) := by
| Mathlib.Topology.Inseparable.503_0.2NeLzt0mQ64QlfB | theorem map_mk_nhdsSet : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
⊢ comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
| Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
| comap mk (𝓝ˢ t) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image] | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) := by
conv_lhs => | Mathlib.Topology.Inseparable.507_0.2NeLzt0mQ64QlfB | theorem comap_mk_nhdsSet : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
x : X
y : Y
⊢ Filter.map (Prod.map mk mk) (𝓝 (x... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq] | theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
| Mathlib.Topology.Inseparable.528_0.2NeLzt0mQ64QlfB | theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s✝ : Set X
f g : X → Y
t s : Set (SeparationQuotient X)
x : X
⊢ Filter.map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s]... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds] | theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x := by
| Mathlib.Topology.Inseparable.533_0.2NeLzt0mQ64QlfB | theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] mk x | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → α
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk] | @[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
| Mathlib.Topology.Inseparable.554_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → α
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk] | @[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
| Mathlib.Topology.Inseparable.560_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
{s : Set (SeparationQuotient X)} {l : Filter α} :
Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage] | @[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
| Mathlib.Topology.Inseparable.580_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
{s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y
hf : ∀ (x y : X), (x ~ᵢ y) → f x = f y
... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ] | @[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f := by
| Mathlib.Topology.Inseparable.586_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
Continuous (lift f hf) ↔ Continuous f | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [← map_prod_map_mk_nhds, tendsto_map'_iff] | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
| Mathlib.Topology.Inseparable.604_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rfl | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
| Mathlib.Topology.Inseparable.604_0.2NeLzt0mQ64QlfB | @[simp]
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal] | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s]... | Mathlib.Topology.Inseparable.612_0.2NeLzt0mQ64QlfB | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s]... | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rfl | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s]... | Mathlib.Topology.Inseparable.612_0.2NeLzt0mQ64QlfB | @[simp] theorem tendsto_lift₂_nhdsWithin {f : X → Y → α}
{hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} {x : X} {y : Y}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s]... | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map,
continuousWithinAt_lift₂] | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
| Mathlib.Topology.Inseparable.636_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s✝ : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d ... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | rfl | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) := by
simp_rw [ContinuousOn, (surjective_mk.Prod_map surje... | Mathlib.Topology.Inseparable.636_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)} :
ContinuousOn (uncurry <| lift₂ f hf) s ↔ ContinuousOn (uncurry f) (Prod.map mk mk ⁻¹' s) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f✝ g : X → Y
t : Set (SeparationQuotient X)
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d :... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp only [continuous_iff_continuousOn_univ, continuousOn_lift₂, preimage_univ] | @[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
| Mathlib.Topology.Inseparable.645_0.2NeLzt0mQ64QlfB | @[simp]
theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d} :
Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
h : ∀ (x : X), f x ~ᵢ g x
⊢ Continuous f ↔ Continu... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)] | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
| Mathlib.Topology.Inseparable.653_0.2NeLzt0mQ64QlfB | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g | Mathlib_Topology_Inseparable |
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
h : ∀ (x : X), f x ~ᵢ g x
⊢ Continuous (Separation... | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | exact continuous_congr fun x ↦ SeparationQuotient.mk_eq_mk.mpr (h x) | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g := by
simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)]
| Mathlib.Topology.Inseparable.653_0.2NeLzt0mQ64QlfB | theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
Continuous f ↔ Continuous g | Mathlib_Topology_Inseparable |
x : ℂ
⊢ Real.sin (arg x) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | unfold arg | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ Real.sin
(if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then arcsin ((-x).im / abs x) + π else arcsin ((-x).im / abs x) - π) =
x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | split_ifs | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; | Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
h✝ : 0 ≤ x.re
⊢ Real.sin (arcsin (x.im / abs x)) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
h✝¹ : ¬0 ≤ x.re
h✝ : 0 ≤ x.im
⊢ Real.sin (arcsin ((-x).im / abs x) + π) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
h✝¹ : ¬0 ≤ x.re
h✝ : ¬0 ≤ x.im
⊢ Real.sin (arcsin ((-x).im / abs x) - π) = x.im / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by
unfold arg; split_ifs <;>
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.36_0.CflASCTDE9UCom5 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx : x ≠ 0
⊢ Real.cos (arg x) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [arg] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
hx : x ≠ 0
⊢ Real.cos
(if 0 ≤ x.re then arcsin (x.im / abs x)
else if 0 ≤ x.im then arcsin ((-x).im / abs x) + π else arcsin ((-x).im / abs x) - π) =
x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | split_ifs with h₁ h₂ | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : 0 ≤ x.re
⊢ Real.cos (arcsin (x.im / abs x)) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : 0 ≤ x.re
⊢ sqrt (1 - (x.im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | field_simp [Real.sqrt_sq, (abs.pos hx).le, *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : 0 ≤ x.im
⊢ Real.cos (arcsin ((-x).im / abs x) + π) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Real.cos_add_pi, Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : 0 ≤ x.im
⊢ -sqrt (1 - ((-x).im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : ¬0 ≤ x.im
⊢ Real.cos (arcsin ((-x).im / abs x) - π) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Real.cos_sub_pi, Real.cos_arcsin] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
x : ℂ
hx : x ≠ 0
h₁ : ¬0 ≤ x.re
h₂ : ¬0 ≤ x.im
⊢ -sqrt (1 - ((-x).im / abs x) ^ 2) = x.re / abs x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *] | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (n... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.43_0.CflASCTDE9UCom5 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases eq_or_ne x 0 with (rfl | hx) | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
⊢ ↑(abs 0) * cexp (↑(arg 0) * I) = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
x : ℂ
hx : x ≠ 0
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have : abs x ≠ 0 := abs.ne_zero hx | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ ↑(abs x) * cexp (↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | apply Complex.ext | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.a
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ (↑(abs x) * cexp (↑(arg x) * I)).re = x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.a
x : ℂ
hx : x ≠ 0
this : abs x ≠ 0
⊢ (↑(abs x) * cexp (↑(arg x) * I)).im = x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> | Mathlib.Analysis.SpecialFunctions.Complex.Arg.56_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ ↑(abs x) * (cos ↑(arg x) + sin ↑(arg x) * I) = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← exp_mul_I, abs_mul_exp_arg_mul_I] | @[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.65_0.CflASCTDE9UCom5 | @[simp]
theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ abs x * Real.cos (arg x) = x.re | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) | @[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.71_0.CflASCTDE9UCom5 | @[simp]
lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x : ℂ
⊢ abs x * Real.sin (arg x) = x.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) | @[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.75_0.CflASCTDE9UCom5 | @[simp]
lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ abs z = 1 ↔ ∃ θ, cexp (↑θ * I) = z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
z : ℂ
hz : abs z = 1
⊢ cexp (↑(arg z) * I) = z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hz : abs z = 1
⊢ cexp (↑(arg z) * I) = ↑(abs z) * cexp (↑(arg z) * I) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [hz, ofReal_one, one_mul] | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
z : ℂ
⊢ (∃ θ, cexp (↑θ * I) = z) → abs z = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rintro ⟨θ, rfl⟩ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· | Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2.intro
θ : ℝ
⊢ abs (cexp (↑θ * I)) = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact Complex.abs_exp_ofReal_mul_I θ | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine' ⟨fun hz => ⟨arg z, _⟩, _⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.79_0.CflASCTDE9UCom5 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ (Set.range fun x => cexp (↑x * I)) = Metric.sphere 0 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | ext x | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.89_0.CflASCTDE9UCom5 | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case h
x : ℂ
⊢ (x ∈ Set.range fun x => cexp (↑x * I)) ↔ x ∈ Metric.sphere 0 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by
ext x
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.89_0.CflASCTDE9UCom5 | @[simp]
theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) = θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ (if 0 ≤ (↑r * (cos ↑θ + sin ↑θ * I)).re then arcsin ((↑r * (cos ↑θ + sin ↑θ * I)).im / r)
else
if 0 ≤ (↑r * (cos ↑θ + sin ↑θ * I)).im then arcsin ((-(↑r * (cos ↑θ + sin ↑θ * I))).im / r) + π
else arcsin ((-(↑r * (cos ↑θ + sin ↑θ * I))).im / r) - π) =
... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [if_pos] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
⊢ arcsin (Real.sin θ) = θ
case pos.hc r : ℝ hr : 0 < r θ : ℝ hθ : θ ∈ Set.Ioc (-π) π h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) ⊢ 0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ ∉ Set.Icc (-(π / 2)) (π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ < -(π / 2) ∨ π / 2 < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | cases' h₁ with h₁ h₁ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : θ < -(π / 2)
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | replace hθ := hθ.1 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have hcos : Real.cos θ < 0 := by
rw [← neg_pos, ← Real.cos_add_pi]
refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ Real.cos θ < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← neg_pos, ← Real.cos_add_pi] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ 0 < Real.cos (θ + π) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_1
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ -(π / 2) < θ + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case refine'_2
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
⊢ θ + π < π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
⊢ θ < 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith;
linarith; exact hsin.not_le; exact hcos.not_le] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hx₁
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ -(π / 2) ≤ θ + π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hx₂
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ θ + π ≤ π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ ¬0 ≤ Real.sin θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact hsin.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inl.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : θ < -(π / 2)
hθ : -π < θ
hcos : Real.cos θ < 0
hsin : Real.sin θ < 0
⊢ ¬0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact hcos.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
h₁ : π / 2 < θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | replace hθ := hθ.2 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
⊢ θ < π + π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
⊢ 0 ≤ θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith;
linarith; exact hsin; exact hcos.not_le] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ (if 0 ≤ Real.cos θ then arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then arcsin (-Real.sin θ) + π else arcsin (-Real.sin θ) - π) =
θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hx₁
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ -(π / 2) ≤ θ - π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hx₂
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ θ - π ≤ π / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | linarith | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ 0 ≤ Real.sin θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact hsin | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case neg.inr.hnc
r : ℝ
hr : 0 < r
θ : ℝ
h₁ : π / 2 < θ
hθ : θ ≤ π
hcos : Real.cos θ < 0
hsin : 0 ≤ Real.sin θ
⊢ ¬0 ≤ Real.cos θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | exact hcos.not_le | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.96_0.CflASCTDE9UCom5 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
θ : ℝ
hθ : θ ∈ Set.Ioc (-π) π
⊢ arg (cos ↑θ + sin ↑θ * I) = θ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.121_0.CflASCTDE9UCom5 | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
⊢ arg 0 = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [arg, le_refl] | @[simp]
theorem arg_zero : arg 0 = 0 := by | Mathlib.Analysis.SpecialFunctions.Complex.Arg.126_0.CflASCTDE9UCom5 | @[simp]
theorem arg_zero : arg 0 = 0 | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
x y : ℂ
h₁ : abs x = abs y
h₂ : arg x = arg y
⊢ x = y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] | theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.130_0.CflASCTDE9UCom5 | theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have hπ : 0 < π := Real.pi_pos | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
hπ : 0 < π
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases eq_or_ne z 0 with (rfl | hz) | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
hπ : 0 < π
⊢ arg 0 ∈ Set.Ioc (-π) π
case inr z : ℂ hπ : 0 < π hz : z ≠ 0 ⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp [hπ, hπ.le] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); | Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr
z : ℂ
hπ : 0 < π
hz : z ≠ 0
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + N • (2 * π) ∈ Set.Ioc (-π) (-π + 2 * π)
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
⊢ arg z ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
this : arg (↑(abs z) * (cos ↑(arg z + ↑N * (2 * π)) + sin ↑(arg z + ↑N * (2 * π)) * I)) = arg z + ↑N * (2 * π)
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | push_cast at this | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inr.intro.intro
z : ℂ
hπ : 0 < π
hz : z ≠ 0
N : ℤ
hN : arg z + ↑N * (2 * π) ∈ Set.Ioc (-π) π
this : arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) = arg z + ↑N * (2 * π)
⊢ arg (↑(abs z) * (cos (↑(arg z) + ↑N * (2 * ↑π)) + sin (↑(arg z) + ↑N * (2 * ↑π)) * I)) ∈ Set.Ioc (-π) π | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rwa [this] | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by
have hπ : 0 < π := Real.pi_pos
rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le]
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN
rw [← abs_mul_cos_... | Mathlib.Analysis.SpecialFunctions.Complex.Arg.138_0.CflASCTDE9UCom5 | theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
z : ℂ
⊢ 0 ≤ arg z ↔ 0 ≤ z.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | rcases eq_or_ne z 0 with (rfl | h₀) | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
| Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
case inl
⊢ 0 ≤ arg 0 ↔ 0 ≤ 0.im | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | simp | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by
rcases eq_or_ne z 0 with (rfl | h₀); · | Mathlib.Analysis.SpecialFunctions.Complex.Arg.166_0.CflASCTDE9UCom5 | @[simp]
theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | Mathlib_Analysis_SpecialFunctions_Complex_Arg |
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