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 |
|---|---|---|---|---|---|---|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | refine' ⟨fun h => _, fun h => _⟩ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [← coe_contLinear_eq_linear, h] | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [← coe_linear_eq_coe_contLinear, h] | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | intro q | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | refine' ⟨fun h => _, fun h => _⟩ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h] | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [← coe_to_affineMap, h] | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGro... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp_rw [h₁, h₂] | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
h... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ | theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | @[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.122_0.bJ3guivW1IqbUMZ | @[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | @[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.122_0.bJ3guivW1IqbUMZ | @[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rcases f with ⟨f, h⟩ | theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.153_0.bJ3guivW1IqbUMZ | theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case mk
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
ins... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def] | theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.153_0.bJ3guivW1IqbUMZ | theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [norm_def] | @[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.181_0.bJ3guivW1IqbUMZ | @[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h, norm_zero] | @[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.181_0.bJ3guivW1IqbUMZ | @[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp [(ContinuousAffineMap.zero_apply)] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp only [coe_add, max_le_iff] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [add_contLinear] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp [(ContinuousAffineMap.neg_apply)] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h₁] at h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h₁] at h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | obtain ⟨q, rfl⟩ := h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp only [norm_eq_zero] at h₁ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [coe_const, Function.const_apply] at h₁ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h₁] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | obtain ⟨q, rfl⟩ := h₁ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp only [norm_le_zero_iff] at h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [coe_const, Function.const_apply] at h₂ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [h₂] | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCom... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rfl | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
s... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ | noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp only [norm_def, (smul_contLinear), norm_smul] | instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.223_0.bJ3guivW1IqbUMZ | instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)] | instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.223_0.bJ3guivW1IqbUMZ | instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [norm_def, max_le_iff] | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | constructor | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
| Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case left
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
i... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.nor... | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [f.decomp] | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case right
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]... | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_a... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [le_add_iff_nonneg_right] | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_a... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | apply norm_nonneg | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_a... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ | theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [f.decomp] | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply] | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rw [ContinuousAffineMap.coe_const, Function.const_apply] | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : Me... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | rintro ⟨v, f⟩ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case mk
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
ins... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | ext | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case mk.a
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
i... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
case mk.a.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂... | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analy... | simp | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.cont... | Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ | /-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f | Mathlib_Analysis_NormedSpace_ContinuousAffineMap |
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G x✝² ⊗[R] M
⊢ ((... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | refine' x.induction_on _ _ _ | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine'_1
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by ... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine'_2
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by ... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine'_3
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by ... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm | /--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
g : G i
m : M
⊢ (toDirectLimit f M) ((of R... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | rw [toDirectLimit, lift.tmul, lift_of] | variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
| Mathlib.Algebra.Module.DirectLimitAndTensorProduct.66_0.NNGkUEQVeDhgQVm | variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
g : G i
m : M
⊢ ((LinearMap.compr₂ (mk R (... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | rfl | variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
| Mathlib.Algebra.Module.DirectLimitAndTensorProduct.66_0.NNGkUEQVeDhgQVm | variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
⊢ Dir... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_ | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
| Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 =>... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | cases isEmpty_or_nonempty ι | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_2
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 =>... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | cases isEmpty_or_nonempty ι | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inl
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | ext | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inl.h
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | apply Subsingleton.elim | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inr
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inr.refine_1
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inr.refine_2
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_1.inr.refine_3
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_2.inl
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | ext | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_2.inl.h
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | apply Subsingleton.elim | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
case refine_2.inr
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by aesop) | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : ... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.e... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm | /--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i j :... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | aesop | /--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) :=
TensorProduct.comm _ _ _ ≪≫ₗ directLimitLeft f M ≪≫ₗ
Module.DirectLimit.congr (fun i ↦ TensorProduct.comm _ _ _)
(fun i j h ↦ TensorP... | Mathlib.Algebra.Module.DirectLimitAndTensorProduct.95_0.NNGkUEQVeDhgQVm | /--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i : ι... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | simp [directLimitRight, congr_apply_of] | @[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g) := by
| Mathlib.Algebra.Module.DirectLimitAndTensorProduct.104_0.NNGkUEQVeDhgQVm | @[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g) | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i : ι... | /-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-l... | simp [directLimitRight, congr_symm_apply_of] | @[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
(directLimitRight f M).symm (of _ _ _ _ _ (m ⊗ₜ g)) = m ⊗ₜ of _ _ _ f _ g := by
| Mathlib.Algebra.Module.DirectLimitAndTensorProduct.108_0.NNGkUEQVeDhgQVm | @[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
(directLimitRight f M).symm (of _ _ _ _ _ (m ⊗ₜ g)) = m ⊗ₜ of _ _ _ f _ g | Mathlib_Algebra_Module_DirectLimitAndTensorProduct |
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ expUnitary (a + b) = expUnitary a * expUnitary b | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | ext | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
| Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b | Mathlib_Analysis_NormedSpace_Star_Exponential |
case a
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ ↑(expUnitary (a + b)) = ↑(expUnitary a * expUnitary b) | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm] | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
| Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b | Mathlib_Analysis_NormedSpace_Star_Exponential |
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ Commute (I • ↑a) (I • ↑b) | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | unfold Commute SemiconjBy | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
| Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b | Mathlib_Analysis_NormedSpace_Star_Exponential |
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ I • ↑a * I • ↑b = I • ↑b * I • ↑a | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm] | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
| Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b | Mathlib_Analysis_NormedSpace_Star_Exponential |
case a
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
hcomm : Commute (I • ↑a) (I • ↑b)
⊢ ↑(expUnitary (a + b)) = ↑(expUnitary a * expUnitary b) | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | simpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
| Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b | Mathlib_Analysis_NormedSpace_Star_Exponential |
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c451... | rw [← h.expUnitary_add, ← h.symm.expUnitary_add, add_comm] | theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by
| Mathlib.Analysis.NormedSpace.Star.Exponential.51_0.QAfFqeOSAt19Prd | theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) | Mathlib_Analysis_NormedSpace_Star_Exponential |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c a : G₀
hc : c ≠ 0
⊢ a / c / (b / c) = a / b | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [div_div_eq_mul_div, div_mul_cancel _ hc] | theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.129_0.ICkvbDoLYHVogsB | theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c a : G₀
hc : c ≠ 0
⊢ a / c * (c / b) = a / b | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [← mul_div_assoc, div_mul_cancel _ hc] | theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.133_0.ICkvbDoLYHVogsB | theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b✝ c a b : G₀
h : b = 0 → a = 0
hb : b = 0
⊢ a / b * b = a | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | simp [*] | theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by | Mathlib.Algebra.GroupWithZero.Units.Lemmas.137_0.ICkvbDoLYHVogsB | theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b✝ c a b : G₀
h : b = 0 → a = 0
hb : b = 0
⊢ a * b / b = a | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | simp [*] | theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by | Mathlib.Algebra.GroupWithZero.Units.Lemmas.141_0.ICkvbDoLYHVogsB | theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c d a b : G₀
h : a = 0 → b = 0
⊢ a * b / a = b | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [mul_comm, mul_div_cancel_of_imp h] | theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.161_0.ICkvbDoLYHVogsB | theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c d a b : G₀
h : b = 0 → a = 0
⊢ b * (a / b) = a | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [mul_comm, div_mul_cancel_of_imp h] | theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.169_0.ICkvbDoLYHVogsB | theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c✝ d✝ a b c d : G₀
hb : b ≠ 0
hd : d ≠ 0
h : a / b = c / d
⊢ a * d = c * b | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [← mul_one a, ← div_self hb, ← mul_comm_div, h, div_mul_eq_mul_div, div_mul_cancel _ hd] | theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.181_0.ICkvbDoLYHVogsB | theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a b✝ c d b : G₀
h : a ≠ 0
⊢ 1 / (a * b) * a = 1 / b | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [div_mul_eq_mul_div, one_mul, div_mul_right _ h] | theorem div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.199_0.ICkvbDoLYHVogsB | theorem div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
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