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 inr.ih.intro
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Infinite F
β α : E
⊢ ∃ α_1, F⟮α_1⟯ = restrictScalars F (↥F⟮α⟯)⟮β⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simp_rw [adjoin_simple_adjoin_simple, eq_comm] | theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
... | Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ | theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K | Mathlib_FieldTheory_PrimitiveElement |
case inr.ih.intro
F : Type u_1
E : Type u_2
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
K : IntermediateField F E
this : FiniteDimensional F E
h✝ : Infinite F
β α : E
⊢ ∃ α_1, F⟮α, β⟯ = F⟮α_1⟯ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact primitive_element_inf_aux_of_finite_intermediateField F α β | theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K := by
haveI := finiteDimensional_of_finite_intermediateField F E
rcases finite_or_infinite F with (_ | _)
· obtain ⟨α, h⟩ := exists_primitive_element_of_finite_bot F K
... | Mathlib.FieldTheory.PrimitiveElement.294_0.R5HND7n71i1v1rZ | theorem exists_primitive_element_of_finite_intermediateField
[Finite (IntermediateField F E)] (K : IntermediateField F E) : ∃ α : E, F⟮α⟯ = K | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
h : ∃ α, F⟮α⟯ = ⊤
⊢ FiniteDimensional F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨α, hprim⟩ := h | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E := by
| Mathlib.FieldTheory.PrimitiveElement.305_0.R5HND7n71i1v1rZ | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
α : E
hprim : F⟮α⟯ = ⊤
⊢ FiniteDimensional F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have hfin := adjoin.finiteDimensional (halg α).isIntegral | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E := by
obtain ⟨α, hprim⟩ := h
| Mathlib.FieldTheory.PrimitiveElement.305_0.R5HND7n71i1v1rZ | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
α : E
hprim : F⟮α⟯ = ⊤
hfin : FiniteDimensional F ↥F⟮α⟯
⊢ FiniteDimensional F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [hprim] at hfin | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E := by
obtain ⟨α, hprim⟩ := h
have hfin := adjoin.finiteDimensional (halg α).isIntegral
| Mathlib.FieldTheory.PrimitiveElement.305_0.R5HND7n71i1v1rZ | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
α : E
hprim : F⟮α⟯ = ⊤
hfin : FiniteDimensional F ↥⊤
⊢ FiniteDimensional F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact topEquiv.toLinearEquiv.finiteDimensional | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E := by
obtain ⟨α, hprim⟩ := h
have hfin := adjoin.finiteDimensional (halg α).isIntegral
rw [hprim] at hfin
| Mathlib.FieldTheory.PrimitiveElement.305_0.R5HND7n71i1v1rZ | theorem finiteDimensional_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : FiniteDimensional F E | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
h : ∃ α, F⟮α⟯ = ⊤
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | haveI := finiteDimensional_of_exists_primitive_element F E halg h | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
| Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
h : ∃ α, F⟮α⟯ = ⊤
this : FiniteDimensional F E
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | obtain ⟨α, hprim⟩ := h | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
| Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let f : F[X] := minpoly F α | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
| Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let G := { g : E[X] // g.Monic ∧ g ∣ f.map (algebraMap F E) } | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have hfin : Finite G := @Finite.of_fintype _ <| fintypeSubtypeMonicDvd
(f.map (algebraMap F E)) <| map_ne_zero (minpoly.ne_zero_of_finite F α) | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
⊢ Finite (IntermediateField F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | let g : IntermediateField F E → G := fun K ↦
⟨(minpoly K α).map (algebraMap K E), (minpoly.monic <| .of_finite K α).map _, by
convert Polynomial.map_dvd (algebraMap K E) (minpoly.dvd_map_of_isScalarTower F K α)
rw [Polynomial.map_map]; rfl⟩ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
K : IntermediateField F E
⊢ Polynomial.map (... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | convert Polynomial.map_dvd (algebraMap K E) (minpoly.dvd_map_of_isScalarTower F K α) | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case h.e'_4
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
K : IntermediateField F E
⊢ Poly... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [Polynomial.map_map] | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case h.e'_4
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
K : IntermediateField F E
⊢ Poly... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rfl | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G :=
... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | have hinj : Function.Injective g := fun K K' heq ↦ by
rw [Subtype.mk.injEq] at heq
apply_fun fun f : E[X] ↦ adjoin F (f.frange : Set E) at heq
simpa only [adjoin_minpoly_coeff_of_exists_primitive_element F hprim] using heq | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G :=
fun K =>
... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [Subtype.mk.injEq] at heq | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G :=
fun K =>
... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | apply_fun fun f : E[X] ↦ adjoin F (f.frange : Set E) at heq | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G :=
fun K =>
... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simpa only [adjoin_minpoly_coeff_of_exists_primitive_element F hprim] using heq | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
case intro
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
halg : Algebra.IsAlgebraic F E
this : FiniteDimensional F E
α : E
hprim : F⟮α⟯ = ⊤
f : F[X] := minpoly F α
G : Type u_2 := { g // Monic g ∧ g ∣ Polynomial.map (algebraMap F E) f }
hfin : Finite G
g : IntermediateField F E → G :=
... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact Finite.of_injective g hinj | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) := by
haveI := finiteDimensional_of_exists_primitive_element F E halg h
obtain ⟨α, hprim⟩ := h
-- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f... | Mathlib.FieldTheory.PrimitiveElement.313_0.R5HND7n71i1v1rZ | theorem finite_intermediateField_of_exists_primitive_element (halg : Algebra.IsAlgebraic F E)
(h : ∃ α : E, F⟮α⟯ = ⊤) : Finite (IntermediateField F E) | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
inst✝⁴ : FiniteDimensional F E
inst✝³ : IsSeparable F E
K : Type u_3
inst✝² : Field K
inst✝¹ : IsAlgClosed K
inst✝ : Algebra F K
⊢ Fintype.card (E →ₐ[F] K) = finrank F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | convert (AlgHom.card_of_powerBasis (L := K) (Field.powerBasisOfFiniteOfSeparable F E)
(IsSeparable.separable _ _) (IsAlgClosed.splits_codomain _)).trans (PowerBasis.finrank _).symm | @[simp]
theorem AlgHom.card (K : Type*) [Field K] [IsAlgClosed K] [Algebra F K] :
Fintype.card (E →ₐ[F] K) = finrank F E := by
| Mathlib.FieldTheory.PrimitiveElement.352_0.R5HND7n71i1v1rZ | @[simp]
theorem AlgHom.card (K : Type*) [Field K] [IsAlgClosed K] [Algebra F K] :
Fintype.card (E →ₐ[F] K) = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra F L
hL : ∀ (x : E), Splits (algebraMap F L) (minpoly F x)
⊢ Fintype.card (E →ₐ[F] L) = finrank F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [← Fintype.ofEquiv_card <| Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits
(AlgebraicClosure L) (Algebra.IsAlgebraic.of_finite F E) _ hL] | @[simp]
theorem AlgHom.card_of_splits (L : Type*) [Field L] [Algebra F L]
(hL : ∀ x : E, (minpoly F x).Splits (algebraMap F L)) :
Fintype.card (E →ₐ[F] L) = finrank F E := by
| Mathlib.FieldTheory.PrimitiveElement.359_0.R5HND7n71i1v1rZ | @[simp]
theorem AlgHom.card_of_splits (L : Type*) [Field L] [Algebra F L]
(hL : ∀ x : E, (minpoly F x).Splits (algebraMap F L)) :
Fintype.card (E →ₐ[F] L) = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F : Type u_1
E : Type u_2
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra F L
hL : ∀ (x : E), Splits (algebraMap F L) (minpoly F x)
⊢ Fintype.card (E →ₐ[F] AlgebraicClosure L) = finrank F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | convert AlgHom.card F E (AlgebraicClosure L) | @[simp]
theorem AlgHom.card_of_splits (L : Type*) [Field L] [Algebra F L]
(hL : ∀ x : E, (minpoly F x).Splits (algebraMap F L)) :
Fintype.card (E →ₐ[F] L) = finrank F E := by
rw [← Fintype.ofEquiv_card <| Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits
(AlgebraicClosure L) (Algebra.IsAlgebraic.of_finite F E... | Mathlib.FieldTheory.PrimitiveElement.359_0.R5HND7n71i1v1rZ | @[simp]
theorem AlgHom.card_of_splits (L : Type*) [Field L] [Algebra F L]
(hL : ∀ x : E, (minpoly F x).Splits (algebraMap F L)) :
Fintype.card (E →ₐ[F] L) = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝⁸ : Field F✝
inst✝⁷ : Field E✝
inst✝⁶ : Algebra F✝ E✝
inst✝⁵ : FiniteDimensional F✝ E✝
inst✝⁴ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ F⟮α⟯ = ⊤ ↔ natDegree (minpoly F α) = finrank F E | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [← adjoin.finrank (IsIntegral.of_finite F α), ← finrank_top F E] | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E := by
| Mathlib.FieldTheory.PrimitiveElement.375_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝⁸ : Field F✝
inst✝⁷ : Field E✝
inst✝⁶ : Algebra F✝ E✝
inst✝⁵ : FiniteDimensional F✝ E✝
inst✝⁴ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ F⟮α⟯ = ⊤ ↔ finrank F ↥F⟮α⟯ = finrank F ↥⊤ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | refine ⟨fun h => ?_, fun h => eq_of_le_of_finrank_eq le_top h⟩ | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E := by
rw [← adjoin.finrank (IsIntegral.of_finite F α), ← finrank_top F E]
| Mathlib.FieldTheory.PrimitiveElement.375_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝⁸ : Field F✝
inst✝⁷ : Field E✝
inst✝⁶ : Algebra F✝ E✝
inst✝⁵ : FiniteDimensional F✝ E✝
inst✝⁴ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
h : F⟮α⟯ = ⊤
⊢ finrank F ↥F⟮α⟯ = finrank F ↥⊤ | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact congr_arg (fun K : IntermediateField F E => finrank F K) h | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E := by
rw [← adjoin.finrank (IsIntegral.of_finite F α), ← finrank_top F E]
refine ⟨fun h => ?_, fun h => eq_of_le_of_finrank_eq le_top h⟩
| Mathlib.FieldTheory.PrimitiveElement.375_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝⁸ : Field F✝
inst✝⁷ : Field E✝
inst✝⁶ : Algebra F✝ E✝
inst✝⁵ : FiniteDimensional F✝ E✝
inst✝⁴ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ F⟮α⟯ = ⊤ ↔ degree (minpoly F α) = ↑(finrank F E) | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [degree_eq_iff_natDegree_eq, primitive_element_iff_minpoly_natDegree_eq] | theorem primitive_element_iff_minpoly_degree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).degree = finrank F E := by
| Mathlib.FieldTheory.PrimitiveElement.381_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_minpoly_degree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).degree = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝⁸ : Field F✝
inst✝⁷ : Field E✝
inst✝⁶ : Algebra F✝ E✝
inst✝⁵ : FiniteDimensional F✝ E✝
inst✝⁴ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
α : E
⊢ minpoly F α ≠ 0 | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact minpoly.ne_zero_of_finite F α | theorem primitive_element_iff_minpoly_degree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).degree = finrank F E := by
rw [degree_eq_iff_natDegree_eq, primitive_element_iff_minpoly_natDegree_eq]
| Mathlib.FieldTheory.PrimitiveElement.381_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_minpoly_degree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).degree = finrank F E | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | classical
simp_rw [primitive_element_iff_minpoly_natDegree_eq, ← card_rootSet_eq_natDegree (K := A)
(IsSeparable.separable F α) (hA _), ← toFinset_card,
← (Algebra.IsAlgebraic.of_finite F E).range_eval_eq_rootSet_minpoly_of_splits _ hA α,
← AlgHom.card_of_splits F E A hA, Fintype.card, toFinset_range, Fin... | theorem primitive_element_iff_algHom_eq_of_eval' (α : E) :
F⟮α⟯ = ⊤ ↔ Function.Injective fun φ : E →ₐ[F] A ↦ φ α := by
| Mathlib.FieldTheory.PrimitiveElement.389_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval' (α : E) :
F⟮α⟯ = ⊤ ↔ Function.Injective fun φ : E →ₐ[F] A ↦ φ α | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | simp_rw [primitive_element_iff_minpoly_natDegree_eq, ← card_rootSet_eq_natDegree (K := A)
(IsSeparable.separable F α) (hA _), ← toFinset_card,
← (Algebra.IsAlgebraic.of_finite F E).range_eval_eq_rootSet_minpoly_of_splits _ hA α,
← AlgHom.card_of_splits F E A hA, Fintype.card, toFinset_range, Finset.card_ima... | theorem primitive_element_iff_algHom_eq_of_eval' (α : E) :
F⟮α⟯ = ⊤ ↔ Function.Injective fun φ : E →ₐ[F] A ↦ φ α := by
classical
| Mathlib.FieldTheory.PrimitiveElement.389_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval' (α : E) :
F⟮α⟯ = ⊤ ↔ Function.Injective fun φ : E →ₐ[F] A ↦ φ α | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
| Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebra | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
| Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | haveI := isSeparable_tower_top_of_isSeparable F F⟮α⟯ E | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebr... | Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | rw [IntermediateField.finrank_top, ← AlgHom.card_of_splits _ _ A, Fintype.card_eq_one_iff] | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebr... | Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact ⟨{ __ := φ, commutes' := fun _ ↦ rfl }, fun ψ ↦ AlgHom.restrictScalars_injective F <|
Eq.symm <| h _ (ψ.commutes <| AdjoinSimple.gen F α).symm⟩ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebr... | Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
F✝ : Type u_1
E✝ : Type u_2
inst✝¹¹ : Field F✝
inst✝¹⁰ : Field E✝
inst✝⁹ : Algebra F✝ E✝
inst✝⁸ : FiniteDimensional F✝ E✝
inst✝⁷ : IsSeparable F✝ E✝
F : Type u_3
E : Type u_4
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
inst✝³ : FiniteDimensional F E
inst✝² : IsSeparable F E
A : Type u_5
inst✝¹ : Field A
inst... | /-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.IntegralDomai... | exact fun x ↦ (IsIntegral.of_finite F x).minpoly_splits_tower_top (hA x) | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ := by
refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩
letI : Algebra F⟮α⟯ A := (φ.comp F⟮α⟯.val).toAlgebr... | Mathlib.FieldTheory.PrimitiveElement.398_0.R5HND7n71i1v1rZ | theorem primitive_element_iff_algHom_eq_of_eval (α : E)
(φ : E →ₐ[F] A) : F⟮α⟯ = ⊤ ↔ ∀ ψ : E →ₐ[F] A, φ α = ψ α → φ = ψ | Mathlib_FieldTheory_PrimitiveElement |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
X Y : J
f : X ⟶ Y
⊢ ((Functor.const J).obj (of ↑{u | ∀ {i j : J} (f : i ⟶ j), (F.map f) (u i) = u j})).map f ≫
(fun j => ContinuousMap.mk fun u => ↑u j) Y =
(fun j => ContinuousMap.mk fun u => ↑u j) X ≫ F.map f | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f ... | Mathlib.Topology.Category.TopCat.Limits.Basic.46_0.lEtiPsL1dQihFhl | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
X Y : J
f : X ⟶ Y
⊢ (𝟙 (of { x // ∀ {i j : J} (f : i ⟶ j), (F.map f) (x i) = x j }) ≫ ContinuousMap.mk fun u => ↑u Y) =
(ContinuousMap.mk fun u => ↑u X) ≫ F.map f | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [Category.id_comp] | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f ... | Mathlib.Topology.Category.TopCat.Limits.Basic.46_0.lEtiPsL1dQihFhl | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
X Y : J
f : X ⟶ Y
⊢ (ContinuousMap.mk fun u => ↑u Y) = (ContinuousMap.mk fun u => ↑u X) ≫ F.map f | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | apply ContinuousMap.ext | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f ... | Mathlib.Topology.Category.TopCat.Limits.Basic.46_0.lEtiPsL1dQihFhl | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
X Y : J
f : X ⟶ Y
⊢ ∀ (a : { x // ∀ {i j : J} (f : i ⟶ j), (F.map f) (x i) = x j }),
(ContinuousMap.mk fun u => ↑u Y) a = ((ContinuousMap.mk fun u => ↑u X) ≫ F.map f) a | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | intro a | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f ... | Mathlib.Topology.Category.TopCat.Limits.Basic.46_0.lEtiPsL1dQihFhl | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
X Y : J
f : X ⟶ Y
a : { x // ∀ {i j : J} (f : i ⟶ j), (F.map f) (x i) = x j }
⊢ (ContinuousMap.mk fun u => ↑u Y) a = ((ContinuousMap.mk fun u => ↑u X) ≫ F.map f) a | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | exact (a.2 f).symm | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt := TopCat.of { u : ∀ j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f ... | Mathlib.Topology.Category.TopCat.Limits.Basic.46_0.lEtiPsL1dQihFhl | /-- A choice of limit cone for a functor `F : J ⥤ TopCat`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `Types.limitCone`).
-/
def limitCone (F : J ⥤ TopCatMax.{v, u}) : Cone F where
pt | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i✝ j✝ : J
f : i✝ ⟶ j✝
⊢ (F.map f) ((fun j => (S.π.app j) x) i✝) = (fun j => (S.π.app j) x) j✝ | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i✝ j✝ : J
f : i✝ ⟶ j✝
⊢ (F.map f) ((S.π.app i✝) x) = (S.π.app j✝) x | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | erw [← S.w f] | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i✝ j✝ : J
f : i✝ ⟶ j✝
⊢ (F.map f) ((S.π.app i✝) x) = (S.π.app i✝ ≫ F.map f) x | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rfl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i j : J
f : i ⟶ j
⊢ (F.map f) ((fun j => (S.π.app j) x) i) = (fun j => (S.π.app j) x) j | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i j : J
f : i ⟶ j
⊢ (F.map f) ((S.π.app i) x) = (S.π.app j) x | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [← S.w f] | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
x : ↑S.pt
i j : J
f : i ⟶ j
⊢ (F.map f) ((S.π.app i) x) = (S.π.app i ≫ F.map f) x | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rfl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
⊢ m =
(fun S =>
ContinuousMap.mk fun x =>
{ val := fun j => (S.π.app j) x,
property :=
(_ : ∀ {i j : J} (f : i ⟶ j), (F.map f) ... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | apply ContinuousMap.ext | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
⊢ ∀ (a : ↑S.pt),
m a =
((fun S =>
ContinuousMap.mk fun x =>
{ val := fun j => (S.π.app j) x,
property :=
... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | intros a | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
a : ↑S.pt
⊢ m a =
((fun S =>
ContinuousMap.mk fun x =>
{ val := fun j => (S.π.app j) x,
property :=
(_ : ∀ {i j ... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | apply Subtype.ext | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h.a
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
a : ↑S.pt
⊢ ↑(m a) =
↑(((fun S =>
ContinuousMap.mk fun x =>
{ val := fun j => (S.π.app j) x,
property :=
... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | funext j | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h.a.h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
a : ↑S.pt
j : J
⊢ ↑(m a) j =
↑(((fun S =>
ContinuousMap.mk fun x =>
{ val := fun j => (S.π.app j) x,
property :=... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h.a.h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
a : ↑S.pt
j : J
⊢ ↑(m a) j = (S.π.app j) a | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [← h] | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case h.a.h
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
S : Cone F
m : S.pt ⟶ (limitCone F).pt
h : ∀ (j : J), m ≫ (limitCone F).π.app j = S.π.app j
a : ↑S.pt
j : J
⊢ ↑(m a) j = (m ≫ (limitCone F).π.app j) a | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rfl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib.Topology.Category.TopCat.Limits.Basic.82_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitCone.{v,u} F) where
lift S... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
⊢ IsLimit (limitConeInfi F) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | refine IsLimit.ofFaithful forget (Types.limitConeIsLimit.{v,u} (F ⋙ forget))
-- Porting note: previously could infer all ?_ except continuity
(fun s => ⟨fun v => ⟨ fun j => (Functor.mapCone forget s).π.app j v, ?_⟩, ?_⟩) fun s => ?_ | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
v : ↑s.pt
⊢ (fun j => (forget.mapCone s).π.app j v) ∈ Functor.sections (F ⋙ forget) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp [Functor.sections] | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
v : ↑s.pt
⊢ ∀ {j j' : J} (f : j ⟶ j'), forget.map (F.map f) (forget.map (s.π.app j) v) = forget.map (s.π.app j') v | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | intro _ _ _ | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
v : ↑s.pt
j✝ j'✝ : J
f✝ : j✝ ⟶ j'✝
⊢ forget.map (F.map f✝) (forget.map (s.π.app j✝) v) = forget.map (s.π.app j'✝) v | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [← comp_apply', forget_map_eq_coe, ← s.π.naturality, forget_map_eq_coe] | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
v : ↑s.pt
j✝ j'✝ : J
f✝ : j✝ ⟶ j'✝
⊢ (((Functor.const J).obj s.pt).map f✝ ≫ s.π.app j'✝) v = (s.π.app j'✝) v | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
v : ↑s.pt
j✝ j'✝ : J
f✝ : j✝ ⟶ j'✝
⊢ (𝟙 s.pt ≫ s.π.app j'✝) v = (s.π.app j'✝) v | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [Category.id_comp] | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_2
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
⊢ Continuous fun v =>
{ val := fun j => (forget.mapCone s).π.app j v,
property := (_ : (fun j => (forget.mapCone s).π.app j v) ∈ Functor.sections (F ⋙ forget)) } | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | exact
continuous_iff_coinduced_le.mpr
(le_iInf fun j =>
coinduced_le_iff_le_induced.mp <|
(continuous_iff_coinduced_le.mp (s.π.app j).continuous : _)) | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_3
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cone F
⊢ forget.map
((fun s =>
ContinuousMap.mk fun v =>
{ val := fun j => (forget.mapCone s).π.app j v,
property := (_ : (fun j => (forget.mapCone s).π.app j v) ∈ Functor.sections (F ⋙ forget)) })
s... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rfl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) :=... | Mathlib.Topology.Category.TopCat.Limits.Basic.105_0.lEtiPsL1dQihFhl | /-- The chosen cone `TopCat.limitConeInfi F` for a functor `F : J ⥤ TopCat` is a limit cone.
Generally you should just use `limit.isLimit F`, unless you need the actual definition
(which is in terms of `Types.limitConeIsLimit`).
-/
def limitConeInfiIsLimit (F : J ⥤ TopCatMax.{v, u}) : IsLimit (limitConeInfi.{v,u} F) | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
⊢ IsColimit (colimitCocone F) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | refine
IsColimit.ofFaithful forget (Types.colimitCoconeIsColimit _) (fun s =>
-- Porting note: it appears notation for forget breaks dot notation (also above)
-- Porting note: previously function was inferred
⟨Quot.lift (fun p => (Functor.mapCocone forget s).ι.app p.fst p.snd) ?_, ?_⟩) fun s => ?_ | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
⊢ ∀ (a b : (j : J) × (F ⋙ forget).obj j),
Types.Quot.Rel (F ⋙ forget) a b →
(fun p => (forget.mapCocone s).ι.app p.fst p.snd) a = (fun p => (forget.mapCocone s).ι.app p.fst p.snd) b | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | intro _ _ ⟨_, h⟩ | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
a✝ b✝ : (j : J) × (F ⋙ forget).obj j
w✝ : a✝.fst ⟶ b✝.fst
h : b✝.snd = (F ⋙ forget).map w✝ a✝.snd
⊢ (fun p => (forget.mapCocone s).ι.app p.fst p.snd) a✝ = (fun p => (forget.mapCocone s).ι.app p.fst p.snd) b✝ | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
a✝ b✝ : (j : J) × (F ⋙ forget).obj j
w✝ : a✝.fst ⟶ b✝.fst
h : b✝.snd = (F ⋙ forget).map w✝ a✝.snd
⊢ forget.map (s.ι.app a✝.fst) a✝.snd = forget.map (s.ι.app b✝.fst) b✝.snd | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [h, Functor.comp_map, ← comp_apply', s.ι.naturality] | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
a✝ b✝ : (j : J) × (F ⋙ forget).obj j
w✝ : a✝.fst ⟶ b✝.fst
h : b✝.snd = (F ⋙ forget).map w✝ a✝.snd
⊢ forget.map (s.ι.app a✝.fst) a✝.snd = forget.map (s.ι.app a✝.fst ≫ ((Functor.const J).obj s.pt).map w✝) a✝.snd | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | dsimp | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_1
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
a✝ b✝ : (j : J) × (F ⋙ forget).obj j
w✝ : a✝.fst ⟶ b✝.fst
h : b✝.snd = (F ⋙ forget).map w✝ a✝.snd
⊢ forget.map (s.ι.app a✝.fst) a✝.snd = forget.map (s.ι.app a✝.fst ≫ 𝟙 s.pt) a✝.snd | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rw [Category.comp_id] | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_2
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
⊢ Continuous
(Quot.lift (fun p => (forget.mapCocone s).ι.app p.fst p.snd)
(_ :
∀ (a b : (j : J) × (F ⋙ forget).obj j),
Types.Quot.Rel (F ⋙ forget) a b →
(fun p => (forget.mapCocone s).ι.app p.fst p.sn... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | exact
continuous_iff_le_induced.mpr
(iSup_le fun j =>
coinduced_le_iff_le_induced.mp <|
(continuous_iff_coinduced_le.mp (s.ι.app j).continuous : _)) | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
case refine_3
J : Type v
inst✝ : SmallCategory J
F : J ⥤ TopCatMax
s : Cocone F
⊢ forget.map
((fun s =>
ContinuousMap.mk
(Quot.lift (fun p => (forget.mapCocone s).ι.app p.fst p.snd)
(_ :
∀ (a b : (j : J) × (F ⋙ forget).obj j),
Types.Quot.Rel (F... | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | rfl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib.Topology.Category.TopCat.Limits.Basic.166_0.lEtiPsL1dQihFhl | /-- The chosen cocone `TopCat.colimitCocone F` for a functor `F : J ⥤ TopCat` is a colimit cocone.
Generally you should just use `colimit.isColimit F`, unless you need the actual definition
(which is in terms of `Types.colimitCoconeIsColimit`).
-/
def colimitCoconeIsColimit (F : J ⥤ TopCatMax.{v, u}) : IsColimit (colim... | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
X : TopCat
⊢ Continuous fun x => PUnit.unit | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | continuity | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) :=
haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X =>
⟨⟨⟨fun _ => PUnit.unit, by | Mathlib.Topology.Category.TopCat.Limits.Basic.213_0.lEtiPsL1dQihFhl | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
X : TopCat
f : X ⟶ of PUnit.{u + 1}
⊢ f = default | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | ext | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) :=
haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X =>
⟨⟨⟨fun _ => PUnit.unit, by continuity⟩⟩, fun f => by | Mathlib.Topology.Category.TopCat.Limits.Basic.213_0.lEtiPsL1dQihFhl | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) | Mathlib_Topology_Category_TopCat_Limits_Basic |
case w
J : Type v
inst✝ : SmallCategory J
X : TopCat
f : X ⟶ of PUnit.{u + 1}
x✝ : forget.obj X
⊢ f x✝ = default x✝ | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | aesop | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) :=
haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X =>
⟨⟨⟨fun _ => PUnit.unit, by continuity⟩⟩, fun f => by ext; | Mathlib.Topology.Category.TopCat.Limits.Basic.213_0.lEtiPsL1dQihFhl | /-- The terminal object of `Top` is `PUnit`. -/
def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
X : TopCat
⊢ Continuous fun x => PEmpty.elim x | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | continuity | /-- The initial object of `Top` is `PEmpty`. -/
def isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) :=
haveI : ∀ X, Unique (TopCat.of PEmpty.{u + 1} ⟶ X) := fun X =>
⟨⟨⟨fun x => x.elim, by | Mathlib.Topology.Category.TopCat.Limits.Basic.225_0.lEtiPsL1dQihFhl | /-- The initial object of `Top` is `PEmpty`. -/
def isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) | Mathlib_Topology_Category_TopCat_Limits_Basic |
J : Type v
inst✝ : SmallCategory J
X : TopCat
f : of PEmpty.{u + 1} ⟶ X
⊢ f = default | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.CategoryTheory.Limits.ConcreteCategory
#align_import topolo... | ext ⟨⟩ | /-- The initial object of `Top` is `PEmpty`. -/
def isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) :=
haveI : ∀ X, Unique (TopCat.of PEmpty.{u + 1} ⟶ X) := fun X =>
⟨⟨⟨fun x => x.elim, by continuity⟩⟩, fun f => by | Mathlib.Topology.Category.TopCat.Limits.Basic.225_0.lEtiPsL1dQihFhl | /-- The initial object of `Top` is `PEmpty`. -/
def isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) | Mathlib_Topology_Category_TopCat_Limits_Basic |
B : Type u
inst✝ : Quiver B
a b : B
f g : Discrete (Path a b)
η : f ⟶ g
⊢ PrelaxFunctor.map₂ (preinclusion B) η = eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map g) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rcases η with ⟨⟨⟩⟩ | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) := by
| Mathlib.CategoryTheory.Bicategory.Coherence.93_0.scNCB7gGNV3iY0Z | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) | Mathlib_CategoryTheory_Bicategory_Coherence |
case up.up
B : Type u
inst✝ : Quiver B
a b : B
f g : Discrete (Path a b)
down✝ : f.as = g.as
⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =
eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map g) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | cases Discrete.ext _ _ (by assumption) | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) := by
rcases η with ⟨⟨⟩⟩
| Mathlib.CategoryTheory.Bicategory.Coherence.93_0.scNCB7gGNV3iY0Z | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b : B
f g : Discrete (Path a b)
down✝ : f.as = g.as
⊢ ?m.4739.as = ?m.4740.as | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | assumption | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) := by
rcases η with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by | Mathlib.CategoryTheory.Bicategory.Coherence.93_0.scNCB7gGNV3iY0Z | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) | Mathlib_CategoryTheory_Bicategory_Coherence |
case up.up.refl
B : Type u
inst✝ : Quiver B
a b : B
f : Discrete (Path a b)
down✝ : f.as = f.as
⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =
eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map f) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | convert (inclusionPath a b).map_id _ | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) := by
rcases η with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
| Mathlib.CategoryTheory.Bicategory.Coherence.93_0.scNCB7gGNV3iY0Z | @[simp]
theorem preinclusion_map₂ {a b : B} (f g : Discrete (Path.{v + 1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom η))) | Mathlib_CategoryTheory_Bicategory_Coherence |
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η : f ⟶ g
⊢ normalizeAux p f = normalizeAux p g | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rcases η with ⟨η'⟩ | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
| Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η : f ⟶ g
η' : Hom₂ f g
⊢ normalizeAux p f = normalizeAux p g | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | apply @congr_fun _ _ fun p => normalizeAux p f | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
| Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h
B : Type u
inst✝ : Quiver B
a b c : B
p : Path a b
f g : Hom b c
η : f ⟶ g
η' : Hom₂ f g
⊢ (fun p => normalizeAux p f) = fun p => normalizeAux p g | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | clear p η | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
η' : Hom₂ f g
⊢ (fun p => normalizeAux p f) = fun p => normalizeAux p g | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction η' with
| vcomp _ _ _ _ => apply Eq.trans <;> assumption
| whisker_left _ _ ih => funext; apply congr_fun ih
| whisker_right _ _ ih => funext; apply congr_arg₂ _ (congr_fun ih _) rfl
| _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
η' : Hom₂ f g
⊢ (fun p => normalizeAux p f) = fun p => normalizeAux p g | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | induction η' with
| vcomp _ _ _ _ => apply Eq.trans <;> assumption
| whisker_left _ _ ih => funext; apply congr_fun ih
| whisker_right _ _ ih => funext; apply congr_arg₂ _ (congr_fun ih _) rfl
| _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : Hom₂ f✝ g✝
θ✝ : Hom₂ g✝ h✝
η_ih✝ : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
θ_ih✝ : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p f✝) = fun p => no... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | vcomp _ _ _ _ => apply Eq.trans <;> assumption | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.vcomp
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : Hom₂ f✝ g✝
θ✝ : Hom₂ g✝ h✝
η_ih✝ : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
θ_ih✝ : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p f✝) = fun p => no... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | apply Eq.trans | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.vcomp.h₁
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : Hom₂ f✝ g✝
θ✝ : Hom₂ g✝ h✝
η_ih✝ : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
θ_ih✝ : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p f✝) = ?mk.h.vc... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | assumption | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.vcomp.h₂
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ g✝ h✝ : a✝ ⟶ b✝
η✝ : Hom₂ f✝ g✝
θ✝ : Hom₂ g✝ h✝
η_ih✝ : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
θ_ih✝ : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p g✝) = fun p =>... | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | assumption | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p (f✝ ≫ g✝)) = fun p => normalizeAux p (f✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | whisker_left _ _ ih => funext; apply congr_fun ih | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_left
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
⊢ (fun p => normalizeAux p (f✝ ≫ g✝)) = fun p => normalizeAux p (f✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_left.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ h✝ : b✝ ⟶ c✝
η✝ : Hom₂ g✝ h✝
ih : (fun p => normalizeAux p g✝) = fun p => normalizeAux p h✝
x✝ : Path a a✝
⊢ normalizeAux x✝ (f✝ ≫ g✝) = normalizeAux x✝ (f✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | apply congr_fun ih | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h✝ : b✝ ⟶ c✝
η✝ : Hom₂ f✝ g✝
ih : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
⊢ (fun p => normalizeAux p (Hom.comp f✝ h✝)) = fun p => normalizeAux p (Hom.comp g✝ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | whisker_right _ _ ih => funext; apply congr_arg₂ _ (congr_fun ih _) rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_right
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h✝ : b✝ ⟶ c✝
η✝ : Hom₂ f✝ g✝
ih : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
⊢ (fun p => normalizeAux p (Hom.comp f✝ h✝)) = fun p => normalizeAux p (Hom.comp g✝ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.whisker_right.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ : FreeBicategory B
f✝ g✝ : a✝ ⟶ b✝
h✝ : b✝ ⟶ c✝
η✝ : Hom₂ f✝ g✝
ih : (fun p => normalizeAux p f✝) = fun p => normalizeAux p g✝
x✝ : Path a a✝
⊢ normalizeAux x✝ (Hom.comp f✝ h✝) = normalizeAux x✝ (Hom.comp g✝ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | apply congr_arg₂ _ (congr_fun ih _) rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.id
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.id
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
⊢ (fun p => normalizeAux p f✝) = fun p => normalizeAux p f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.id.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
x✝ : Path a a✝
⊢ normalizeAux x✝ f✝ = normalizeAux x✝ f✝ | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.associator
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
⊢ (fun p => normalizeAux p ((f✝ ≫ g✝) ≫ h✝)) = fun p => normalizeAux p (f✝ ≫ g✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | | _ => funext; rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.associator
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
⊢ (fun p => normalizeAux p ((f✝ ≫ g✝) ≫ h✝)) = fun p => normalizeAux p (f✝ ≫ g✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | funext | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
case mk.h.associator.h
B : Type u
inst✝ : Quiver B
a b c : B
f g : Hom b c
a✝ b✝ c✝ d✝ : FreeBicategory B
f✝ : a✝ ⟶ b✝
g✝ : b✝ ⟶ c✝
h✝ : c✝ ⟶ d✝
x✝ : Path a a✝
⊢ normalizeAux x✝ ((f✝ ≫ g✝) ≫ h✝) = normalizeAux x✝ (f✝ ≫ g✝ ≫ h✝) | /-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.Categ... | rfl | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normali... | Mathlib.CategoryTheory.Bicategory.Coherence.145_0.scNCB7gGNV3iY0Z | /-- Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalizeAux p f = normalizeAux p g`.
-/
theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g | Mathlib_CategoryTheory_Bicategory_Coherence |
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