state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case inl.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ≥0
⊢ singularPart (↑r • s) μ = ↑r • singularPart s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact singularPart_smul_nnreal s μ r | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ singularPart (r • s) μ = r • singularPart s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | | inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
congr
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul]
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul... | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ singularPart (r • s) μ = r • singularPart s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [singularPart, singularPart] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) -
toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) =
r •
(t... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | conv_lhs =>
congr
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul]
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) -
toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul]
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) -
toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul]
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) -
toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ)
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure ... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | · congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).posPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | · rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).posPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | · congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| toSignedMeasure (Measure.singularPart (toJordanDecomposition (r • s)).negPart μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | congr | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | · rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case a.μ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
| Measure.singularPart (toJordanDecomposition (r • s)).negPart μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_negPart_neg _ _ hr, singularPart_smul] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
r : ℝ
hr : r < 0
⊢ toSignedMeasure (Real.toNNReal (-r) • Measure.singularPart (toJordanDecomposition s).negPart μ) -
toSignedMeasure (Real.toNNReal (-r) • Measure.singularPart (toJordanDecomposition s)... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [toSignedMeasure_smul, toSignedMeasure_smul, ← neg_sub, ← smul_sub, NNReal.smul_def,
← neg_smul, Real.coe_toNNReal _ (le_of_lt (neg_pos.mpr hr)), neg_neg] | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ := by
cases le_or_lt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
con... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.363_0.HPGboz0rhL6sBes | nonrec theorem singularPart_smul (s : SignedMeasure α) (μ : Measure α) (r : ℝ) :
(r • s).singularPart μ = r • s.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
⊢ singularPart (s + t) μ = singularPart s μ + singularPart t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine'
(eq_singularPart _ (s.rnDeriv μ + t.rnDeriv μ)
((mutuallySingular_singularPart s μ).add_left (mutuallySingular_singularPart t μ))
_).symm | theorem singularPart_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s + t).singularPart μ = s.singularPart μ + t.singularPart μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.383_0.HPGboz0rhL6sBes | theorem singularPart_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s + t).singularPart μ = s.singularPart μ + t.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
⊢ s + t = singularPart s μ + singularPart t μ + withDensityᵥ μ (rnDeriv s μ + rnDeriv t μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [withDensityᵥ_add (integrable_rnDeriv s μ) (integrable_rnDeriv t μ), add_assoc,
add_comm (t.singularPart μ), add_assoc, add_comm _ (t.singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, ← add_assoc,
singularPart_add_withDensity_rnDeriv_eq] | theorem singularPart_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s + t).singularPart μ = s.singularPart μ + t.singularPart μ := by
refine'
(eq_singularPart _ (s.rnDeriv μ + t.rnDeriv μ)
((mutuallySingular_singularPart s μ).add_left... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.383_0.HPGboz0rhL6sBes | theorem singularPart_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s + t).singularPart μ = s.singularPart μ + t.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
⊢ singularPart (s - t) μ = singularPart s μ - singularPart t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [sub_eq_add_neg, sub_eq_add_neg, singularPart_add, singularPart_neg] | theorem singularPart_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s - t).singularPart μ = s.singularPart μ - t.singularPart μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.396_0.HPGboz0rhL6sBes | theorem singularPart_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] :
(s - t).singularPart μ = s.singularPart μ - t.singularPart μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
⊢ f =ᶠ[ae μ] rnDeriv s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | set f' := hfi.1.mk f | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
⊢ f =ᶠ[ae μ] rnDeriv s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | have hadd' : s = t + μ.withDensityᵥ f' := by
convert hadd using 2
exact WithDensityᵥEq.congr_ae hfi.1.ae_eq_mk.symm | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
⊢ s = t + withDensityᵥ μ f' | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | convert hadd using 2 | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.e'_3.h.e'_6
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
⊢ withDensityᵥ μ f' = withDensityᵥ μ f | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact WithDensityᵥEq.congr_ae hfi.1.ae_eq_mk.symm | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
hadd' : s = t + withDensityᵥ μ f'
⊢ f =ᶠ[ae μ] rnDeriv s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | have := haveLebesgueDecomposition_mk μ hfi.1.measurable_mk htμ hadd' | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
hadd' : s = t + withDensityᵥ μ f'
this : HaveLebesgueDecomposit... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine' (Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) hfi _).symm | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t✝ t : SignedMeasure α
f : α → ℝ
hfi : Integrable f
htμ : t ⟂ᵥ toENNRealVectorMeasure μ
hadd : s = t + withDensityᵥ μ f
f' : α → ℝ := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)
hadd' : s = t + withDensityᵥ μ f'
this : HaveLebesgueDecomposit... | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← add_right_inj t, ← hadd, eq_singularPart _ f htμ hadd,
singularPart_add_withDensity_rnDeriv_eq] | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.402_0.HPGboz0rhL6sBes | /-- Given a measure `μ`, signed measures `s` and `t`, and a function `f` such that `t` is
mutually singular with respect to `μ` and `s = t + μ.withDensityᵥ f`, we have
`f = rnDeriv s μ`, i.e. `f` is the Radon-Nikodym derivative of `s` and `μ`. -/
theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f ... | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ rnDeriv (-s) μ =ᶠ[ae μ] -rnDeriv s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg _ | theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.418_0.HPGboz0rhL6sBes | theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
⊢ withDensityᵥ μ (rnDeriv (-s) μ) = withDensityᵥ μ (-rnDeriv s μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [withDensityᵥ_neg, ← add_right_inj ((-s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_neg, ← neg_add,
singularPart_add_withDensity_rnDeriv_eq] | theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ := by
refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg _
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.418_0.HPGboz0rhL6sBes | theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
⊢ rnDeriv (r • s) μ =ᶠ[ae μ] r • rnDeriv s μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)
((integrable_rnDeriv _ _).smul r) _ | theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
(r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.427_0.HPGboz0rhL6sBes | theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
(r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t s : SignedMeasure α
μ : Measure α
inst✝ : HaveLebesgueDecomposition s μ
r : ℝ
⊢ withDensityᵥ μ (rnDeriv (r • s) μ) = withDensityᵥ μ (r • rnDeriv s μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [withDensityᵥ_smul (rnDeriv s μ) r, ← add_right_inj ((r • s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_smul, ← smul_add,
singularPart_add_withDensity_rnDeriv_eq] | theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
(r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ := by
refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)
((integrable_rnDeriv _ _).smul r) _
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.427_0.HPGboz0rhL6sBes | theorem rnDeriv_smul (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] (r : ℝ) :
(r • s).rnDeriv μ =ᵐ[μ] r • s.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝² : HaveLebesgueDecomposition s μ
inst✝¹ : HaveLebesgueDecomposition t μ
inst✝ : HaveLebesgueDecomposition (s + t) μ
⊢ rnDeriv (s + t) μ =ᶠ[ae μ] rnDeriv s μ + rnDeriv t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)
((integrable_rnDeriv _ _).add (integrable_rnDeriv _ _)) _ | theorem rnDeriv_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
(s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.437_0.HPGboz0rhL6sBes | theorem rnDeriv_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
(s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝² : HaveLebesgueDecomposition s μ
inst✝¹ : HaveLebesgueDecomposition t μ
inst✝ : HaveLebesgueDecomposition (s + t) μ
⊢ withDensityᵥ μ (rnDeriv (s + t) μ) = withDensityᵥ μ (rnDeriv s μ + rnDeriv t μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← add_right_inj ((s + t).singularPart μ), singularPart_add_withDensity_rnDeriv_eq,
withDensityᵥ_add (integrable_rnDeriv _ _) (integrable_rnDeriv _ _), singularPart_add,
add_assoc, add_comm (t.singularPart μ), add_assoc, add_comm _ (t.singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, ← add_assoc,... | theorem rnDeriv_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
(s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ := by
refine'
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _)
((integrabl... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.437_0.HPGboz0rhL6sBes | theorem rnDeriv_add (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [(s + t).HaveLebesgueDecomposition μ] :
(s + t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ + t.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
hst : HaveLebesgueDecomposition (s - t) μ
⊢ rnDeriv (s - t) μ =ᶠ[ae μ] rnDeriv s μ - rnDeriv t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [sub_eq_add_neg] at hst | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.450_0.HPGboz0rhL6sBes | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
hst : HaveLebesgueDecomposition (s + -t) μ
⊢ rnDeriv (s - t) μ =ᶠ[ae μ] rnDeriv s μ - rnDeriv t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [sub_eq_add_neg, sub_eq_add_neg] | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ := by
rw [sub_eq_add_neg] at hst
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.450_0.HPGboz0rhL6sBes | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
s✝ t✝ s t : SignedMeasure α
μ : Measure α
inst✝¹ : HaveLebesgueDecomposition s μ
inst✝ : HaveLebesgueDecomposition t μ
hst : HaveLebesgueDecomposition (s + -t) μ
⊢ rnDeriv (s + -t) μ =ᶠ[ae μ] rnDeriv s μ + -rnDeriv t μ | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact ae_eq_trans (rnDeriv_add _ _ _) (Filter.EventuallyEq.add (ae_eq_refl _) (rnDeriv_neg _ _)) | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ := by
rw [sub_eq_add_neg] at hst
rw [sub_eq_add_neg, sub_eq_add_neg]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.450_0.HPGboz0rhL6sBes | theorem rnDeriv_sub (s t : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ]
[t.HaveLebesgueDecomposition μ] [hst : (s - t).HaveLebesgueDecomposition μ] :
(s - t).rnDeriv μ =ᵐ[μ] s.rnDeriv μ - t.rnDeriv μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
c✝ c : ComplexMeasure α
μ : Measure α
⊢ Integrable (rnDeriv c μ) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← memℒp_one_iff_integrable, ← memℒp_re_im_iff] | theorem integrable_rnDeriv (c : ComplexMeasure α) (μ : Measure α) : Integrable (c.rnDeriv μ) μ := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.489_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (c : ComplexMeasure α) (μ : Measure α) : Integrable (c.rnDeriv μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν : Measure α
c✝ c : ComplexMeasure α
μ : Measure α
⊢ Memℒp (fun x => IsROrC.re (rnDeriv c μ x)) 1 ∧ Memℒp (fun x => IsROrC.im (rnDeriv c μ x)) 1 | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact
⟨memℒp_one_iff_integrable.2 (SignedMeasure.integrable_rnDeriv _ _),
memℒp_one_iff_integrable.2 (SignedMeasure.integrable_rnDeriv _ _)⟩ | theorem integrable_rnDeriv (c : ComplexMeasure α) (μ : Measure α) : Integrable (c.rnDeriv μ) μ := by
rw [← memℒp_one_iff_integrable, ← memℒp_re_im_iff]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.489_0.HPGboz0rhL6sBes | theorem integrable_rnDeriv (c : ComplexMeasure α) (μ : Measure α) : Integrable (c.rnDeriv μ) μ | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
⊢ singularPart c μ + Measure.withDensityᵥ μ (rnDeriv c μ) = c | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
| c | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← c.toComplexMeasure_to_signedMeasure] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
| c | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← c.toComplexMeasure_to_signedMeasure] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
| c | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [← c.toComplexMeasure_to_signedMeasure] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
⊢ singularPart c μ + Measure.withDensityᵥ μ (rnDeriv c μ) = SignedMeasure.toComplexMeasure (re c) (im c) | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | ext i hi : 1 | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ ↑(singularPart c μ + Measure.withDensityᵥ μ (rnDeriv c μ)) i = ↑(SignedMeasure.toComplexMeasure (re c) (im c)) i | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ ↑(singularPart c μ) i + ↑(Measure.withDensityᵥ μ (rnDeriv c μ)) i = { re := ↑(re c) i, im := ↑(im c) i } | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | apply Complex.ext | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
| Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ (↑(singularPart c μ) i + ↑(Measure.withDensityᵥ μ (rnDeriv c μ)) i).re = { re := ↑(re c) i, im := ↑(im c) i }.re | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [Complex.add_re, withDensityᵥ_apply (c.integrable_rnDeriv μ) hi, ← IsROrC.re_eq_complex_re,
← integral_re (c.integrable_rnDeriv μ).integrableOn, IsROrC.re_eq_complex_re,
← withDensityᵥ_apply _ hi] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ (↑(singularPart c μ) i).re + ↑(Measure.withDensityᵥ μ fun a => (rnDeriv c μ a).re) i =
{ re := ↑(re c) i, im := ↑(im c) i }.re | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | change (c.re.singularPart μ + μ.withDensityᵥ (c.re.rnDeriv μ)) i = _ | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ ↑(SignedMeasure.singularPart (re c) μ + Measure.withDensityᵥ μ (SignedMeasure.rnDeriv (re c) μ)) i =
{ re := ↑(re c) i, im := ↑(im c) i }.re | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [c.re.singularPart_add_withDensity_rnDeriv_eq μ] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ Integrable fun a => (rnDeriv c μ a).re | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact SignedMeasure.integrable_rnDeriv _ _ | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ (↑(singularPart c μ) i + ↑(Measure.withDensityᵥ μ (rnDeriv c μ)) i).im = { re := ↑(re c) i, im := ↑(im c) i }.im | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [Complex.add_im, withDensityᵥ_apply (c.integrable_rnDeriv μ) hi, ← IsROrC.im_eq_complex_im,
← integral_im (c.integrable_rnDeriv μ).integrableOn, IsROrC.im_eq_complex_im,
← withDensityᵥ_apply _ hi] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ (↑(singularPart c μ) i).im + ↑(Measure.withDensityᵥ μ fun a => (rnDeriv c μ a).im) i =
{ re := ↑(re c) i, im := ↑(im c) i }.im | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | change (c.im.singularPart μ + μ.withDensityᵥ (c.im.rnDeriv μ)) i = _ | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
case h.a
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ ↑(SignedMeasure.singularPart (im c) μ + Measure.withDensityᵥ μ (SignedMeasure.rnDeriv (im c) μ)) i =
{ re := ↑(re c) i, im := ↑(im c) i }.im | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | rw [c.im.singularPart_add_withDensity_rnDeriv_eq μ] | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
c : ComplexMeasure α
inst✝ : HaveLebesgueDecomposition c μ
i : Set α
hi : MeasurableSet i
⊢ Integrable fun a => (rnDeriv c μ a).im | /-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
/-!
# Lebesgue decomposition
This file proves the Lebesgue decomposition ... | exact SignedMeasure.integrable_rnDeriv _ _ | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c := by
conv_rhs => rw [← c.toComplexMeasure_to_signedMeasure]
ext i hi : 1
rw [VectorMeasure.add_apply, SignedMeasure.toComplexMeasure_apply]
apply Complex.ext
· rw [Complex... | Mathlib.MeasureTheory.Decomposition.SignedLebesgue.496_0.HPGboz0rhL6sBes | theorem singularPart_add_withDensity_rnDeriv_eq [c.HaveLebesgueDecomposition μ] :
c.singularPart μ + μ.withDensityᵥ (c.rnDeriv μ) = c | Mathlib_MeasureTheory_Decomposition_SignedLebesgue |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
b : B
⊢ Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b = ContinuousLinearEquiv.refl 𝕜 F | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | ext v | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F := by
| Mathlib.Topology.VectorBundle.Constructions.48_0.ZrgS90NPsSlDzPQ | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F | Mathlib_Topology_VectorBundle_Constructions |
case h.h
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
b : B
v : F
⊢ (Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b) v = (ContinuousLinearEquiv.refl 𝕜 F) v | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [Trivialization.coordChangeL_apply'] | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F := by
ext v
| Mathlib.Topology.VectorBundle.Constructions.48_0.ZrgS90NPsSlDzPQ | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F | Mathlib_Topology_VectorBundle_Constructions |
case h.h
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
b : B
v : F
⊢ (↑(trivialization B F) (↑(PartialHomeomorph.symm (trivialization B F).toPartialHomeomorph) (b, v))).2 =
(ContinuousLinearEquiv.refl 𝕜... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exacts [rfl, ⟨mem_univ _, mem_univ _⟩] | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F := by
ext v
rw [Trivialization.coordChangeL_apply']
| Mathlib.Topology.VectorBundle.Constructions.48_0.ZrgS90NPsSlDzPQ | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
⊢ Trivialization.IsLinear 𝕜 e | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [eq_trivialization B F e] | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
| Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
e : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
⊢ Trivialization.IsLinear 𝕜 (trivialization B F) | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | infer_instance | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
| Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
e e' : Trivialization F TotalSpace.proj
he : MemTrivializationAtlas e
he' : MemTrivializationAtlas e'
⊢ ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜 e... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | obtain rfl := eq_trivialization B F e | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
infer_instance
continuousOn_coordChange' e e' he he' := by
| Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
e' : Trivialization F TotalSpace.proj
he' : MemTrivializationAtlas e'
he : MemTrivializationAtlas (trivialization B F)
⊢ ContinuousOn (fun b => ↑(Trivialization.... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | obtain rfl := eq_trivialization B F e' | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
infer_instance
continuousOn_coordChange' e e' he he' := by
obtain rfl := eq_trivialization B F e
| Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
he he' : MemTrivializationAtlas (trivialization B F)
⊢ ContinuousOn (fun b => ↑(Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b))
... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | simp only [trivialization.coordChangeL] | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
infer_instance
continuousOn_coordChange' e e' he he' := by
obtain rfl := eq_trivialization B F e
obtain rfl := eq_trivialization B F e'
| Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
he he' : MemTrivializationAtlas (trivialization B F)
⊢ ContinuousOn (fun b => ↑(ContinuousLinearEquiv.refl 𝕜 F))
((trivialization B F).baseSet ∩ (trivializa... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exact continuous_const.continuousOn | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he := by
rw [eq_trivialization B F e]
infer_instance
continuousOn_coordChange' e e' he he' := by
obtain rfl := eq_trivialization B F e
obtain rfl := eq_trivialization B F e'
simp only [trivialization.co... | Mathlib.Topology.VectorBundle.Constructions.59_0.ZrgS90NPsSlDzPQ | instance vectorBundle : VectorBundle 𝕜 F (Bundle.Trivial B F) where
trivialization_linear' e he | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap'] | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib.Topology.VectorBundle.Constructions.93_0.ZrgS90NPsSlDzPQ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro ⟨v₁, v₂⟩ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib.Topology.VectorBundle.Constructions.93_0.ZrgS90NPsSlDzPQ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib_Topology_VectorBundle_Constructions |
case mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B →... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | show
(e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂) | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib.Topology.VectorBundle.Constructions.93_0.ZrgS90NPsSlDzPQ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib_Topology_VectorBundle_Constructions |
case mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B →... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply'] | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib.Topology.VectorBundle.Constructions.93_0.ZrgS90NPsSlDzPQ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib_Topology_VectorBundle_Constructions |
case mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B →... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exacts [rfl, hb, ⟨hb.1.2, hb.2.2⟩, ⟨hb.1.1, hb.2.1⟩] | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib.Topology.VectorBundle.Constructions.93_0.ZrgS90NPsSlDzPQ | @[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
(hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
(e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChang... | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
| Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : N... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | skip | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : N... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | infer_instance | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
| Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro _ _ ⟨e₁, e₂, he₁, he₂, rfl⟩ ⟨e₁', e₂', he₁', he₂', rfl⟩ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
| Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : Normed... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | skip | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : Normed... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | refine' (((continuousOn_coordChange 𝕜 e₁ e₁').mono _).prod_mapL 𝕜
((continuousOn_coordChange 𝕜 e₂ e₂').mono _)).congr _ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | dsimp only [baseSet_prod, mfld_simps] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | dsimp only [baseSet_prod, mfld_simps] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | dsimp only [baseSet_prod, mfld_simps] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_1
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | mfld_set_tac | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | mfld_set_tac | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro b hb | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [ContinuousLinearMap.ext_iff] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro ⟨v₁, v₂⟩ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
ins... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | show (e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) =
(e₁.coordChangeL 𝕜 e₁' b v₁, e₂.coordChangeL 𝕜 e₂' b v₂) | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
ins... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [e₁.coordChangeL_apply e₁', e₂.coordChangeL_apply e₂', (e₁.prod e₂).coordChangeL_apply'] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.intro.intro.mk.intro.intro.intro.intro.refine'_3.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
ins... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exacts [rfl, hb, ⟨hb.1.2, hb.2.2⟩, ⟨hb.1.1, hb.2.1⟩] | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' := by
rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩; skip
infer_instance
continuousOn_coordChange' := by
rint... | Mathlib.Topology.VectorBundle.Constructions.126_0.ZrgS90NPsSlDzPQ | /-- The product of two vector bundles is a vector bundle. -/
instance VectorBundle.prod [VectorBundle 𝕜 F₁ E₁] [VectorBundle 𝕜 F₂ E₂] :
VectorBundle 𝕜 (F₁ × F₂) (E₁ ×ᵇ E₂) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | ext v : 2 | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib.Topology.VectorBundle.Constructions.150_0.ZrgS90NPsSlDzPQ | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib_Topology_VectorBundle_Constructions |
case h.h
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | obtain ⟨v₁, v₂⟩ := v | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib.Topology.VectorBundle.Constructions.150_0.ZrgS90NPsSlDzPQ | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib_Topology_VectorBundle_Constructions |
case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ ... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod] | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib.Topology.VectorBundle.Constructions.150_0.ZrgS90NPsSlDzPQ | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib_Topology_VectorBundle_Constructions |
case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ ... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | exact (congr_arg Prod.snd (prod_apply 𝕜 hx.1 hx.2 v₁ v₂) : _) | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib.Topology.VectorBundle.Constructions.150_0.ZrgS90NPsSlDzPQ | @[simp] -- porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
{e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
(hx : x ∈ (e₁.prod e₂).baseSet) :
(e₁.prod e... | Mathlib_Topology_VectorBundle_Constructions |
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : (x : B) → AddCommMono... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro _ ⟨e, he, rfl⟩ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
| Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : (... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | infer_instance | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
| Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace B
inst✝⁵ : (x : B) → AddCommMono... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
| Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.mk.intro.intro
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpa... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | refine' ((continuousOn_coordChange 𝕜 e e').comp
(map_continuous f).continuousOn fun b hb => hb).congr _ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
| Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
case mk.intro.intro.mk.intro.intro
R : Type u_1
𝕜 : Type u_2
B : Type u_3
F : Type u_4
E : B → Type u_5
B' : Type u_6
f✝ : B' → B
inst✝¹¹ : TopologicalSpace B'
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpa... | /-
Copyright © 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
#align_import to... | rintro b (hb : f b ∈ e.baseSet ∩ e'.baseSet) | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' := by
rintro _ ⟨e, he, rfl⟩
infer_instance
continuousOn_coordChange' := by
rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
refi... | Mathlib.Topology.VectorBundle.Constructions.184_0.ZrgS90NPsSlDzPQ | instance VectorBundle.pullback [∀ x, TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E]
(f : K) : VectorBundle 𝕜 F ((f : B' → B) *ᵖ E) where
trivialization_linear' | Mathlib_Topology_VectorBundle_Constructions |
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