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 this.Hcont
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | apply (H.iteratedFDeriv m).continuousOn.congr | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hcont
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | intro x hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hcont
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x✝ : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact iteratedFDerivWithin_of_isOpen _ t_open hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hdiff
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | rintro m - | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hdiff
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | apply (H.iteratedFDeriv m).differentiableOn.congr | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hdiff
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | intro x hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hdiff
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x✝ : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact iteratedFDerivWithin_of_isOpen _ t_open hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s := by
let t := { x | AnalyticAt 𝕜 f x }
suffices : ContDiffOn 𝕜 n f t; exact this.mono h
have H : AnalyticOn 𝕜 f t := fun x hx => hx
have t_open... | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
ContDiffOn 𝕜 n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : AnalyticAt 𝕜 f x
n : ℕ∞
⊢ ContDif... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | obtain ⟨s, hs, hf⟩ := h.exists_mem_nhds_analyticOn | theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
ContDiffAt 𝕜 n f x := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.161_0.XLJ3uW4JYwyXQcn | theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
ContDiffAt 𝕜 n f x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s✝ : Set E
inst✝ : CompleteSpace F
h : AnalyticAt 𝕜 f ... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact hf.contDiffOn.contDiffAt hs | theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
ContDiffAt 𝕜 n f x := by
obtain ⟨s, hs, hf⟩ := h.exists_mem_nhds_analyticOn
| Mathlib.Analysis.Calculus.FDeriv.Analytic.161_0.XLJ3uW4JYwyXQcn | theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
ContDiffAt 𝕜 n f x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 𝕜 F
r : ℝ≥0∞
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n : ℕ
⊢ Anal... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | induction' n with n IH | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.194_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 𝕜 F
r : ℝ≥0∞
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
⊢ ... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | exact h | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s := by
induction' n with n IH
· | Mathlib.Analysis.Calculus.FDeriv.Analytic.194_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 𝕜 F
r : ℝ≥0∞
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
inst✝ : CompleteSpace F
h : AnalyticOn 𝕜 f s
n ... | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.... | simpa only [Function.iterate_succ', Function.comp_apply] using IH.deriv | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s := by
induction' n with n IH
· exact h
· | Mathlib.Analysis.Calculus.FDeriv.Analytic.194_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
F : Type u
K : Type v
A : Type w
inst✝¹⁰ : CommRing F
inst✝⁹ : Ring K
inst✝⁸ : AddCommGroup A
inst✝⁷ : Algebra F K
inst✝⁶ : Module K A
inst✝⁵ : Module F A
inst✝⁴ : IsScalarTower F K A
inst✝³ : StrongRankCondition F
inst✝² : StrongRankCondition K
inst✝¹ : Module.Free F K
inst✝ : Module.Free K A
⊢ lift.{w, v} (Module.ran... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | obtain ⟨_, b⟩ := Module.Free.exists_basis (R := F) (M := K) | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by... | Mathlib.FieldTheory.Tower.51_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) | Mathlib_FieldTheory_Tower |
case intro.mk
F : Type u
K : Type v
A : Type w
inst✝¹⁰ : CommRing F
inst✝⁹ : Ring K
inst✝⁸ : AddCommGroup A
inst✝⁷ : Algebra F K
inst✝⁶ : Module K A
inst✝⁵ : Module F A
inst✝⁴ : IsScalarTower F K A
inst✝³ : StrongRankCondition F
inst✝² : StrongRankCondition K
inst✝¹ : Module.Free F K
inst✝ : Module.Free K A
fst✝ : Type... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | obtain ⟨_, c⟩ := Module.Free.exists_basis (R := K) (M := A) | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by... | Mathlib.FieldTheory.Tower.51_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) | Mathlib_FieldTheory_Tower |
case intro.mk.intro.mk
F : Type u
K : Type v
A : Type w
inst✝¹⁰ : CommRing F
inst✝⁹ : Ring K
inst✝⁸ : AddCommGroup A
inst✝⁷ : Algebra F K
inst✝⁶ : Module K A
inst✝⁵ : Module F A
inst✝⁴ : IsScalarTower F K A
inst✝³ : StrongRankCondition F
inst✝² : StrongRankCondition K
inst✝¹ : Module.Free F K
inst✝ : Module.Free K A
fs... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank, ←
lift_umax.{w, v}, ← (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift,
lift_lift, lift_umax.{v, w}] | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by... | Mathlib.FieldTheory.Tower.51_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) | Mathlib_FieldTheory_Tower |
F✝ : Type u
K✝ : Type v
A✝ : Type w
inst✝²¹ : CommRing F✝
inst✝²⁰ : Ring K✝
inst✝¹⁹ : AddCommGroup A✝
inst✝¹⁸ : Algebra F✝ K✝
inst✝¹⁷ : Module K✝ A✝
inst✝¹⁶ : Module F✝ A✝
inst✝¹⁵ : IsScalarTower F✝ K✝ A✝
inst✝¹⁴ : StrongRankCondition F✝
inst✝¹³ : StrongRankCondition K✝
inst✝¹² : Module.Free F✝ K✝
inst✝¹¹ : Module.Free... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | convert lift_rank_mul_lift_rank F K A | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib.FieldTheory.Tower.65_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib_FieldTheory_Tower |
case h.e'_2.h.e'_5
F✝ : Type u
K✝ : Type v
A✝ : Type w
inst✝²¹ : CommRing F✝
inst✝²⁰ : Ring K✝
inst✝¹⁹ : AddCommGroup A✝
inst✝¹⁸ : Algebra F✝ K✝
inst✝¹⁷ : Module K✝ A✝
inst✝¹⁶ : Module F✝ A✝
inst✝¹⁵ : IsScalarTower F✝ K✝ A✝
inst✝¹⁴ : StrongRankCondition F✝
inst✝¹³ : StrongRankCondition K✝
inst✝¹² : Module.Free F✝ K✝
in... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [lift_id] | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib.FieldTheory.Tower.65_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib_FieldTheory_Tower |
case h.e'_2.h.e'_6
F✝ : Type u
K✝ : Type v
A✝ : Type w
inst✝²¹ : CommRing F✝
inst✝²⁰ : Ring K✝
inst✝¹⁹ : AddCommGroup A✝
inst✝¹⁸ : Algebra F✝ K✝
inst✝¹⁷ : Module K✝ A✝
inst✝¹⁶ : Module F✝ A✝
inst✝¹⁵ : IsScalarTower F✝ K✝ A✝
inst✝¹⁴ : StrongRankCondition F✝
inst✝¹³ : StrongRankCondition K✝
inst✝¹² : Module.Free F✝ K✝
in... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [lift_id] | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib.FieldTheory.Tower.65_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib_FieldTheory_Tower |
case h.e'_3
F✝ : Type u
K✝ : Type v
A✝ : Type w
inst✝²¹ : CommRing F✝
inst✝²⁰ : Ring K✝
inst✝¹⁹ : AddCommGroup A✝
inst✝¹⁸ : Algebra F✝ K✝
inst✝¹⁷ : Module K✝ A✝
inst✝¹⁶ : Module F✝ A✝
inst✝¹⁵ : IsScalarTower F✝ K✝ A✝
inst✝¹⁴ : StrongRankCondition F✝
inst✝¹³ : StrongRankCondition K✝
inst✝¹² : Module.Free F✝ K✝
inst✝¹¹ :... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [lift_id] | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib.FieldTheory.Tower.65_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.
This is a simpler version of `lift_rank_mul_lift_rank` with `K` and `A` in the same universe. -/
theorem rank_mul_rank (F : Type u) (K A : Type v) [CommRing F] [Ri... | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝¹² : CommRing F
inst✝¹¹ : Ring K
inst✝¹⁰ : AddCommGroup A
inst✝⁹ : Algebra F K
inst✝⁸ : Module K A
inst✝⁷ : Module F A
inst✝⁶ : IsScalarTower F K A
inst✝⁵ : StrongRankCondition F
inst✝⁴ : StrongRankCondition K
inst✝³ : Module.Free F K
inst✝² : Module.Free K A
inst✝¹ : Module.Finite... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | letI := nontrivial_of_invariantBasisNumber F | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A := by
| Mathlib.FieldTheory.Tower.76_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝¹² : CommRing F
inst✝¹¹ : Ring K
inst✝¹⁰ : AddCommGroup A
inst✝⁹ : Algebra F K
inst✝⁸ : Module K A
inst✝⁷ : Module F A
inst✝⁶ : IsScalarTower F K A
inst✝⁵ : StrongRankCondition F
inst✝⁴ : StrongRankCondition K
inst✝³ : Module.Free F K
inst✝² : Module.Free K A
inst✝¹ : Module.Finite... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | let b := Module.Free.chooseBasis F K | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A := by
letI := nontrivial_o... | Mathlib.FieldTheory.Tower.76_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝¹² : CommRing F
inst✝¹¹ : Ring K
inst✝¹⁰ : AddCommGroup A
inst✝⁹ : Algebra F K
inst✝⁸ : Module K A
inst✝⁷ : Module F A
inst✝⁶ : IsScalarTower F K A
inst✝⁵ : StrongRankCondition F
inst✝⁴ : StrongRankCondition K
inst✝³ : Module.Free F K
inst✝² : Module.Free K A
inst✝¹ : Module.Finite... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | let c := Module.Free.chooseBasis K A | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A := by
letI := nontrivial_o... | Mathlib.FieldTheory.Tower.76_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝¹² : CommRing F
inst✝¹¹ : Ring K
inst✝¹⁰ : AddCommGroup A
inst✝⁹ : Algebra F K
inst✝⁸ : Module K A
inst✝⁷ : Module F A
inst✝⁶ : IsScalarTower F K A
inst✝⁵ : StrongRankCondition F
inst✝⁴ : StrongRankCondition K
inst✝³ : Module.Free F K
inst✝² : Module.Free K A
inst✝¹ : Module.Finite... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [finrank_eq_card_basis b, finrank_eq_card_basis c, finrank_eq_card_basis (b.smul c),
Fintype.card_prod] | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A := by
letI := nontrivial_o... | Mathlib.FieldTheory.Tower.76_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
$\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
theorem FiniteDimensional.finrank_mul_finrank' [Module.Finite F K]
[Module.Finite K A] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝⁶ : Field F
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup A
inst✝³ : Algebra F K
inst✝² : Module K A
inst✝¹ : Module F A
inst✝ : IsScalarTower F K A
hf : FiniteDimensional F A
b : Finset A
hb : span F ↑b = ⊤
⊢ restrictScalars F (span K ↑b) = restrictScalars F ⊤ | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [Submodule.restrictScalars_top, eq_top_iff, ← hb, Submodule.span_le] | theorem right [hf : FiniteDimensional F A] : FiniteDimensional K A :=
let ⟨⟨b, hb⟩⟩ := hf
⟨⟨b, Submodule.restrictScalars_injective F _ _ <| by
| Mathlib.FieldTheory.Tower.114_0.ihtkOmbgx804u7P | theorem right [hf : FiniteDimensional F A] : FiniteDimensional K A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝⁶ : Field F
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup A
inst✝³ : Algebra F K
inst✝² : Module K A
inst✝¹ : Module F A
inst✝ : IsScalarTower F K A
hf : FiniteDimensional F A
b : Finset A
hb : span F ↑b = ⊤
⊢ ↑b ⊆ ↑(restrictScalars F (span K ↑b)) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | exact Submodule.subset_span | theorem right [hf : FiniteDimensional F A] : FiniteDimensional K A :=
let ⟨⟨b, hb⟩⟩ := hf
⟨⟨b, Submodule.restrictScalars_injective F _ _ <| by
rw [Submodule.restrictScalars_top, eq_top_iff, ← hb, Submodule.span_le]
| Mathlib.FieldTheory.Tower.114_0.ihtkOmbgx804u7P | theorem right [hf : FiniteDimensional F A] : FiniteDimensional K A | Mathlib_FieldTheory_Tower |
F : Type u
K : Type v
A : Type w
inst✝⁷ : Field F
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup A
inst✝⁴ : Algebra F K
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : FiniteDimensional F K
⊢ finrank F K * finrank K A = finrank F A | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | by_cases hA : FiniteDimensional K A | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A := by
| Mathlib.FieldTheory.Tower.121_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
case pos
F : Type u
K : Type v
A : Type w
inst✝⁷ : Field F
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup A
inst✝⁴ : Algebra F K
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : FiniteDimensional F K
hA : FiniteDimensional K A
⊢ finrank F K * finrank K A = finrank F A | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | replace hA : FiniteDimensional K A := hA | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A := by
by_cases... | Mathlib.FieldTheory.Tower.121_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
case pos
F : Type u
K : Type v
A : Type w
inst✝⁷ : Field F
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup A
inst✝⁴ : Algebra F K
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : FiniteDimensional F K
hA : FiniteDimensional K A
⊢ finrank F K * finrank K A = finrank F A | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [finrank_mul_finrank'] | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A := by
by_cases... | Mathlib.FieldTheory.Tower.121_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
case neg
F : Type u
K : Type v
A : Type w
inst✝⁷ : Field F
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup A
inst✝⁴ : Algebra F K
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : FiniteDimensional F K
hA : ¬FiniteDimensional K A
⊢ finrank F K * finrank K A = finrank F A | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | rw [finrank_of_infinite_dimensional hA, mul_zero, finrank_of_infinite_dimensional] | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A := by
by_cases... | Mathlib.FieldTheory.Tower.121_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
case neg
F : Type u
K : Type v
A : Type w
inst✝⁷ : Field F
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup A
inst✝⁴ : Algebra F K
inst✝³ : Module K A
inst✝² : Module F A
inst✝¹ : IsScalarTower F K A
inst✝ : FiniteDimensional F K
hA : ¬FiniteDimensional K A
⊢ ¬FiniteDimensional F A | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | exact mt (@right F K A _ _ _ _ _ _ _) hA | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A := by
by_cases... | Mathlib.FieldTheory.Tower.121_0.ihtkOmbgx804u7P | /-- Tower law: if `A` is a `K`-vector space and `K` is a field extension of `F` then
`dim_F(A) = dim_F(K) * dim_K(A)`.
This is `FiniteDimensional.finrank_mul_finrank'` with one fewer finiteness assumption. -/
theorem finrank_mul_finrank [FiniteDimensional F K] : finrank F K * finrank K A = finrank F A | Mathlib_FieldTheory_Tower |
F : Type u
K✝ : Type v
A✝ : Type w
inst✝⁹ : Field F
inst✝⁸ : DivisionRing K✝
inst✝⁷ : AddCommGroup A✝
inst✝⁶ : Algebra F K✝
inst✝⁵ : Module K✝ A✝
inst✝⁴ : Module F A✝
inst✝³ : IsScalarTower F K✝ A✝
A : Type u_1
inst✝² : Ring A
inst✝¹ : IsDomain A
inst✝ : Algebra F A
hp : Nat.Prime (finrank F A)
K : Subalgebra F A
⊢ K =... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | haveI : FiniteDimensional _ _ := finiteDimensional_of_finrank hp.pos | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
{ toNontrivial :=
⟨⟨⊥, ⊤, fun he =>
Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
eq_bot_or_eq_top := fun K ... | Mathlib.FieldTheory.Tower.133_0.ihtkOmbgx804u7P | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) | Mathlib_FieldTheory_Tower |
F : Type u
K✝ : Type v
A✝ : Type w
inst✝⁹ : Field F
inst✝⁸ : DivisionRing K✝
inst✝⁷ : AddCommGroup A✝
inst✝⁶ : Algebra F K✝
inst✝⁵ : Module K✝ A✝
inst✝⁴ : Module F A✝
inst✝³ : IsScalarTower F K✝ A✝
A : Type u_1
inst✝² : Ring A
inst✝¹ : IsDomain A
inst✝ : Algebra F A
hp : Nat.Prime (finrank F A)
K : Subalgebra F A
this ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | letI := divisionRingOfFiniteDimensional F K | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
{ toNontrivial :=
⟨⟨⊥, ⊤, fun he =>
Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
eq_bot_or_eq_top := fun K ... | Mathlib.FieldTheory.Tower.133_0.ihtkOmbgx804u7P | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) | Mathlib_FieldTheory_Tower |
F : Type u
K✝ : Type v
A✝ : Type w
inst✝⁹ : Field F
inst✝⁸ : DivisionRing K✝
inst✝⁷ : AddCommGroup A✝
inst✝⁶ : Algebra F K✝
inst✝⁵ : Module K✝ A✝
inst✝⁴ : Module F A✝
inst✝³ : IsScalarTower F K✝ A✝
A : Type u_1
inst✝² : Ring A
inst✝¹ : IsDomain A
inst✝ : Algebra F A
hp : Nat.Prime (finrank F A)
K : Subalgebra F A
this✝... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | refine' (hp.eq_one_or_self_of_dvd _ ⟨_, (finrank_mul_finrank F K A).symm⟩).imp _ fun h => _ | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
{ toNontrivial :=
⟨⟨⊥, ⊤, fun he =>
Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
eq_bot_or_eq_top := fun K ... | Mathlib.FieldTheory.Tower.133_0.ihtkOmbgx804u7P | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) | Mathlib_FieldTheory_Tower |
case refine'_1
F : Type u
K✝ : Type v
A✝ : Type w
inst✝⁹ : Field F
inst✝⁸ : DivisionRing K✝
inst✝⁷ : AddCommGroup A✝
inst✝⁶ : Algebra F K✝
inst✝⁵ : Module K✝ A✝
inst✝⁴ : Module F A✝
inst✝³ : IsScalarTower F K✝ A✝
A : Type u_1
inst✝² : Ring A
inst✝¹ : IsDomain A
inst✝ : Algebra F A
hp : Nat.Prime (finrank F A)
K : Subal... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | exact Subalgebra.eq_bot_of_finrank_one | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
{ toNontrivial :=
⟨⟨⊥, ⊤, fun he =>
Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
eq_bot_or_eq_top := fun K ... | Mathlib.FieldTheory.Tower.133_0.ihtkOmbgx804u7P | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) | Mathlib_FieldTheory_Tower |
case refine'_2
F : Type u
K✝ : Type v
A✝ : Type w
inst✝⁹ : Field F
inst✝⁸ : DivisionRing K✝
inst✝⁷ : AddCommGroup A✝
inst✝⁶ : Algebra F K✝
inst✝⁵ : Module K✝ A✝
inst✝⁴ : Module F A✝
inst✝³ : IsScalarTower F K✝ A✝
A : Type u_1
inst✝² : Ring A
inst✝¹ : IsDomain A
inst✝ : Algebra F A
hp : Nat.Prime (finrank F A)
K : Subal... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Nat.Prime
import Mathlib.RingTheory.AlgebraTower
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
#alig... | exact
Algebra.toSubmodule_eq_top.1 (eq_top_of_finrank_eq <| K.finrank_toSubmodule.trans h) | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
{ toNontrivial :=
⟨⟨⊥, ⊤, fun he =>
Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
eq_bot_or_eq_top := fun K ... | Mathlib.FieldTheory.Tower.133_0.ihtkOmbgx804u7P | theorem Subalgebra.isSimpleOrder_of_finrank_prime (A) [Ring A] [IsDomain A] [Algebra F A]
(hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) | Mathlib_FieldTheory_Tower |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a : k
⊢ slope f a a = 0 | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [slope, sub_self, inv_zero, zero_smul] | @[simp]
theorem slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.46_0.R1BInF4Gl9ffltx | @[simp]
theorem slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
⊢ (b - a) • slope f a b = f b -ᵥ f a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rcases eq_or_ne a b with (rfl | hne) | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.55_0.R1BInF4Gl9ffltx | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a | Mathlib_LinearAlgebra_AffineSpace_Slope |
case inl
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a : k
⊢ (a - a) • slope f a a = f a -ᵥ f a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [sub_self, zero_smul, vsub_self] | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := by
rcases eq_or_ne a b with (rfl | hne)
· | Mathlib.LinearAlgebra.AffineSpace.Slope.55_0.R1BInF4Gl9ffltx | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a | Mathlib_LinearAlgebra_AffineSpace_Slope |
case inr
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
hne : a ≠ b
⊢ (b - a) • slope f a b = f b -ᵥ f a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [slope, smul_inv_smul₀ (sub_ne_zero.2 hne.symm)] | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a := by
rcases eq_or_ne a b with (rfl | hne)
· rw [sub_self, zero_smul, vsub_self]
· | Mathlib.LinearAlgebra.AffineSpace.Slope.55_0.R1BInF4Gl9ffltx | @[simp]
theorem sub_smul_slope (f : k → PE) (a b : k) : (b - a) • slope f a b = f b -ᵥ f a | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
⊢ (b - a) • slope f a b +ᵥ f a = f b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [sub_smul_slope, vsub_vadd] | theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.62_0.R1BInF4Gl9ffltx | theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → E
c : PE
⊢ (slope fun x => f x +ᵥ c) = slope f | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | ext a b | @[simp]
theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.66_0.R1BInF4Gl9ffltx | @[simp]
theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f | Mathlib_LinearAlgebra_AffineSpace_Slope |
case h.h
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → E
c : PE
a b : k
⊢ slope (fun x => f x +ᵥ c) a b = slope f a b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub] | @[simp]
theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f := by
ext a b
| Mathlib.LinearAlgebra.AffineSpace.Slope.66_0.R1BInF4Gl9ffltx | @[simp]
theorem slope_vadd_const (f : k → E) (c : PE) : (slope fun x => f x +ᵥ c) = slope f | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → E
a b : k
h : a ≠ b
⊢ slope (fun x => (x - a) • f x) a b = f b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp [slope, inv_smul_smul₀ (sub_ne_zero.2 h.symm)] | @[simp]
theorem slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b) :
slope (fun x => (x - a) • f x) a b = f b := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.72_0.R1BInF4Gl9ffltx | @[simp]
theorem slope_sub_smul (f : k → E) {a b : k} (h : a ≠ b) :
slope (fun x => (x - a) • f x) a b = f b | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
h : slope f a b = 0
⊢ f a = f b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [← sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd] | theorem eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.78_0.R1BInF4Gl9ffltx | theorem eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝⁶ : Field k
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module k E
inst✝³ : AddTorsor E PE
F : Type u_4
PF : Type u_5
inst✝² : AddCommGroup F
inst✝¹ : Module k F
inst✝ : AddTorsor F PF
f : PE →ᵃ[k] PF
g : k → PE
a b : k
⊢ slope (⇑f ∘ g) a b = f.linear (slope g a b) | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp only [slope, (· ∘ ·), f.linear.map_smul, f.linearMap_vsub] | theorem AffineMap.slope_comp {F PF : Type*} [AddCommGroup F] [Module k F] [AddTorsor F PF]
(f : PE →ᵃ[k] PF) (g : k → PE) (a b : k) : slope (f ∘ g) a b = f.linear (slope g a b) := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.82_0.R1BInF4Gl9ffltx | theorem AffineMap.slope_comp {F PF : Type*} [AddCommGroup F] [Module k F] [AddTorsor F PF]
(f : PE →ᵃ[k] PF) (g : k → PE) (a b : k) : slope (f ∘ g) a b = f.linear (slope g a b) | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
⊢ slope f a b = slope f b a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub] | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.92_0.R1BInF4Gl9ffltx | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | by_cases hab : a = b | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : a = b
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | subst hab | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a c : k
⊢ ((a - a) / (c - a)) • slope f a a + ((c - a) / (c - a)) • slope f a c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [sub_self, zero_div, zero_smul, zero_add] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a c : k
⊢ ((c - a) / (c - a)) • slope f a c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | by_cases hac : a = c | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a c : k
hac : a = c
⊢ ((c - a) / (c - a)) • slope f a c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp [hac] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a c : k
hac : ¬a = c
⊢ ((c - a) / (c - a)) • slope f a c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | by_cases hbc : b = c | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : b = c
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | subst hbc | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case pos
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b : k
hab : ¬a = b
⊢ ((b - a) / (b - a)) • slope f a b + ((b - b) / (b - a)) • slope f b b = slope f a b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp [sub_ne_zero.2 (Ne.symm hab)] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : ¬b = c
⊢ ((b - a) / (c - a)) • slope f a b + ((c - b) / (c - a)) • slope f b c = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [add_comm] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
case neg
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
hab : ¬a = b
hbc : ¬b = c
⊢ ((c - b) / (c - a)) • slope f b c + ((b - a) / (c - a)) • slope f a b = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp_rw [slope, div_eq_inv_mul, mul_smul, ← smul_add,
smul_inv_smul₀ (sub_ne_zero.2 <| Ne.symm hab), smul_inv_smul₀ (sub_ne_zero.2 <| Ne.symm hbc),
vsub_add_vsub_cancel] | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib.LinearAlgebra.AffineSpace.Slope.96_0.R1BInF4Gl9ffltx | /-- `slope f a c` is a linear combination of `slope f a b` and `slope f b c`. This version
explicitly provides coefficients. If `a ≠ c`, then the sum of the coefficients is `1`, so it is
actually an affine combination, see `lineMap_slope_slope_sub_div_sub`. -/
theorem sub_div_sub_smul_slope_add_sub_div_sub_smul_slope (... | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b c : k
h : a ≠ c
⊢ (lineMap (slope f a b) (slope f b c)) ((c - b) / (c - a)) = slope f a c | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | field_simp [sub_ne_zero.2 h.symm, ← sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c,
lineMap_apply_module] | /-- `slope f a c` is an affine combination of `slope f a b` and `slope f b c`. This version uses
`lineMap` to express this property. -/
theorem lineMap_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) :
lineMap (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.116_0.R1BInF4Gl9ffltx | /-- `slope f a c` is an affine combination of `slope f a b` and `slope f b c`. This version uses
`lineMap` to express this property. -/
theorem lineMap_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) :
lineMap (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c | Mathlib_LinearAlgebra_AffineSpace_Slope |
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
⊢ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r = slope f a b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | obtain rfl | hab : a = b ∨ a ≠ b := Classical.em _ | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by
| Mathlib.LinearAlgebra.AffineSpace.Slope.124_0.R1BInF4Gl9ffltx | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b | Mathlib_LinearAlgebra_AffineSpace_Slope |
case inl
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a r : k
⊢ (lineMap (slope f ((lineMap a a) r) a) (slope f a ((lineMap a a) r))) r = slope f a a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | simp | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by
obta... | Mathlib.LinearAlgebra.AffineSpace.Slope.124_0.R1BInF4Gl9ffltx | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b | Mathlib_LinearAlgebra_AffineSpace_Slope |
case inr
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ (lineMap (slope f ((lineMap a b) r) b) (slope f a ((lineMap a b) r))) r = slope f a b | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b] | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by
obta... | Mathlib.LinearAlgebra.AffineSpace.Slope.124_0.R1BInF4Gl9ffltx | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b | Mathlib_LinearAlgebra_AffineSpace_Slope |
case inr
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ (lineMap (slope f b ((lineMap a b) r)) (slope f ((lineMap a b) r) a)) r = slope f b a | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | convert lineMap_slope_slope_sub_div_sub f b (lineMap a b r) a hab.symm using 2 | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by
obta... | Mathlib.LinearAlgebra.AffineSpace.Slope.124_0.R1BInF4Gl9ffltx | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b | Mathlib_LinearAlgebra_AffineSpace_Slope |
case h.e'_2.h.e'_6
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a b r : k
hab : a ≠ b
⊢ r = (a - (lineMap a b) r) / (a - b) | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | rw [lineMap_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, ← sub_sub,
sub_sub_cancel] | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by
obta... | Mathlib.LinearAlgebra.AffineSpace.Slope.124_0.R1BInF4Gl9ffltx | /-- `slope f a b` is an affine combination of `slope f a (lineMap a b r)` and
`slope f (lineMap a b r) b`. We use `lineMap` to express this property. -/
theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b | Mathlib_LinearAlgebra_AffineSpace_Slope |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
⊢ (𝟙 (𝟙_ C) ⊗ 𝟙 (𝟙_ C)) ≫ (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | coherence | /-- The trivial monoid object. We later show this is initial in `Mon_ C`.
-/
@[simps]
def trivial : Mon_ C where
X := 𝟙_ C
one := 𝟙 _
mul := (λ_ _).hom
mul_assoc := by coherence
mul_one := by | Mathlib.CategoryTheory.Monoidal.Mon_.55_0.NTUMzhXPwXsmsYt | /-- The trivial monoid object. We later show this is initial in `Mon_ C`.
-/
@[simps]
def trivial : Mon_ C where
X | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
⊢ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (λ_ (𝟙_ C)).hom =
(α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom ≫ (𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).hom) ≫ (λ_ (𝟙_ C)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | coherence | /-- The trivial monoid object. We later show this is initial in `Mon_ C`.
-/
@[simps]
def trivial : Mon_ C where
X := 𝟙_ C
one := 𝟙 _
mul := (λ_ _).hom
mul_assoc := by | Mathlib.CategoryTheory.Monoidal.Mon_.55_0.NTUMzhXPwXsmsYt | /-- The trivial monoid object. We later show this is initial in `Mon_ C`.
-/
@[simps]
def trivial : Mon_ C where
X | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
Z : C
f : Z ⟶ M.X
⊢ (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← id_tensor_comp_tensor_id, Category.assoc, M.one_mul, leftUnitor_naturality] | @[simp]
theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f := by
| Mathlib.CategoryTheory.Monoidal.Mon_.72_0.NTUMzhXPwXsmsYt | @[simp]
theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
Z : C
f : Z ⟶ M.X
⊢ (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id_comp_id_tensor, Category.assoc, M.mul_one, rightUnitor_naturality] | @[simp]
theorem mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f := by
| Mathlib.CategoryTheory.Monoidal.Mon_.77_0.NTUMzhXPwXsmsYt | @[simp]
theorem mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
⊢ (𝟙 M.X ⊗ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ⊗ 𝟙 M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | theorem assoc_flip :
(𝟙 M.X ⊗ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ⊗ 𝟙 M.X) ≫ M.mul := by | Mathlib.CategoryTheory.Monoidal.Mon_.82_0.NTUMzhXPwXsmsYt | theorem assoc_flip :
(𝟙 M.X ⊗ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ⊗ 𝟙 M.X) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A✝ B✝ : Mon_ C
f : A✝ ⟶ B✝
e : IsIso ((forget C).map f)
⊢ B✝.mul ≫ inv f.hom = (inv f.hom ⊗ inv f.hom) ≫ A✝.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [IsIso.comp_inv_eq, Hom.mul_hom, Category.assoc, ← tensor_comp_assoc,
IsIso.inv_hom_id, tensor_id, Category.id_comp] | /-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/
instance : ReflectsIsomorphisms (forget C) where
reflects f e :=
⟨⟨{ hom := inv f.hom
mul_hom := by
| Mathlib.CategoryTheory.Monoidal.Mon_.152_0.NTUMzhXPwXsmsYt | /-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/
instance : ReflectsIsomorphisms (forget C) where
reflects f e | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A✝ B✝ : Mon_ C
f : A✝ ⟶ B✝
e : IsIso ((forget C).map f)
⊢ f ≫ Hom.mk (inv f.hom) = 𝟙 A✝ ∧ Hom.mk (inv f.hom) ≫ f = 𝟙 B✝ | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/
instance : ReflectsIsomorphisms (forget C) where
reflects f e :=
⟨⟨{ hom := inv f.hom
mul_hom := by
simp only [IsIso.comp_inv_eq, Hom.mul_hom, Category.assoc, ← tensor_comp_assoc,
IsIso.in... | Mathlib.CategoryTheory.Monoidal.Mon_.152_0.NTUMzhXPwXsmsYt | /-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/
instance : ReflectsIsomorphisms (forget C) where
reflects f e | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
⊢ N.one ≫ f.inv = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← one_f] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
⊢ (M.one ≫ f.hom) ≫ f.inv = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
⊢ N.mul ≫ f.inv = (f.inv ⊗ f.inv) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← cancel_mono f.hom] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
⊢ (N.mul ≫ f.inv) ≫ f.hom = ((f.inv ⊗ f.inv) ≫ M.mul) ≫ f.hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 2 3 => rw [mul_f] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
| M.mul ≫ f.hom
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [mul_f] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
| M.mul ≫ f.hom
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [mul_f] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
| M.mul ≫ f.hom
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [mul_f] | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M✝ M N : Mon_ C
f : M.X ≅ N.X
one_f : M.one ≫ f.hom = N.one
mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul
⊢ (N.mul ≫ f.inv) ≫ f.hom = (f.inv ⊗ f.inv) ≫ (f.hom ⊗ f.hom) ≫ N.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib.CategoryTheory.Monoidal.Mon_.161_0.NTUMzhXPwXsmsYt | /-- Construct an isomorphism of monoids by giving an isomorphism between the underlying objects
and checking compatibility with unit and multiplication only in the forward direction.
-/
def isoOfIso {M N : Mon_ C} (f : M.X ≅ N.X) (one_f : M.one ≫ f.hom = N.one)
(mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) : M ... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
⊢ (trivial C).one ≫ A.one = A.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by | Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
⊢ 𝟙 (𝟙_ C) ≫ A.one = A.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; | Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
⊢ (trivial C).mul ≫ A.one = (A.one ⊗ A.one) ≫ A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by | Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
⊢ (λ_ (𝟙_ C)).hom ≫ A.one = (A.one ⊗ A.one) ≫ A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp [A.one_mul, unitors_equal] | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by dsimp; | Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
f : trivial C ⟶ A
⊢ f = default | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | ext | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by dsimp; simp [A.one_mul, unitors_equal] }
uniq f := by
| Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
f : trivial C ⟶ A
⊢ f.hom = default.hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by dsimp; simp [A.one_mul, unitors_equal] }
uniq f := by
ext; | Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
f : trivial C ⟶ A
⊢ f.hom = A.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp f.hom] | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by dsimp; simp [A.one_mul, unitors_equal] }
uniq f := by
ext; simp
| Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M A : Mon_ C
f : trivial C ⟶ A
⊢ 𝟙 (trivial C).X ≫ f.hom = A.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | erw [f.one_hom] | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default :=
{ hom := A.one
one_hom := by dsimp; simp
mul_hom := by dsimp; simp [A.one_mul, unitors_equal] }
uniq f := by
ext; simp
rw [← Category.id_comp f.hom]
| Mathlib.CategoryTheory.Monoidal.Mon_.179_0.NTUMzhXPwXsmsYt | instance uniqueHomFromTrivial (A : Mon_ C) : Unique (trivial C ⟶ A) where
default | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.ε ≫ F.map A.one ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul = (λ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_lhs => rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.ε ≫ F.map A.one ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.ε ≫ F.map A.one ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.ε ≫ F.map A.one ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ ((F.ε ⊗ F.map (𝟙 A.X)) ≫ (F.map A.one ⊗ F.map (𝟙 A.X))) ≫ μ F A.X A.X ≫ F.map A.mul = (λ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.one ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.one ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.one ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.ε ⊗ F.map (𝟙 A.X)) ≫ (μ F (𝟙_ C) A.X ≫ F.map (A.one ⊗ 𝟙 A.X)) ≫ F.map A.mul = (λ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← F.toFunctor.map_comp, A.one_mul] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.one ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.one_mul] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.one ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.one_mul] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.one ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.one_mul] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.ε ⊗ F.map (𝟙 A.X)) ≫ μ F (𝟙_ C) A.X ≫ F.map (λ_ A.X).hom = (λ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.ε ⊗ 𝟙 (F.obj A.X)) ≫ μ F (𝟙_ C) A.X ≫ F.map (λ_ A.X).hom = (λ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.left_unitality] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
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