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 |
|---|---|---|---|---|---|---|
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
⊢ (eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) ((finSuccEquiv R n) f)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{ Polynomial.mapRingHom (eval s) with
commutes' := fun r => by
convert Polynomial.map_C (eval s)
exact (eval_C _).symm } | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
| Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
src✝ : (MvPolynomial (Fin n) R)[X] →+* R[X] := mapRingHom (eval s)
r : R
⊢ OneHom.toFun
(↑↑{ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | convert Polynomial.map_C (eval s) | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
case h.e'_3.h.e'_6
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
src✝ : (MvPolynomial (Fin n) R)[X] →+* R[X] := mapRingHom (eval s)
r : R
⊢ r = (eval s) ((algebraMap R (MvPolynomial (Fin n) R... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact (eval_C _).symm | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
let src := mapRingHom (eval s);
{
toRingHom :=
{ toMonoidHom := ↑src, map_zero' := (_... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | show
aeval (Fin.cons y s : Fin (n + 1) → R) f =
(Polynomial.aeval y).comp (φ.comp (finSuccEquiv R n).toAlgHom) f | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
let src := mapRingHom (eval s);
{
toRingHom :=
{ toMonoidHom := ↑src, map_zero' := (_... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | congr 2 | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
case e_a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
let src := mapRingHom (eval s);
{
toRingHom :=
{ toMonoidHom := ↑src, map_ze... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | apply MvPolynomial.algHom_ext | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
case e_a.hf
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
let src := mapRingHom (eval s);
{
toRingHom :=
{ toMonoidHom := ↑src, map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Fin.forall_fin_succ] | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
case e_a.hf
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s✝ : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s : Fin n → R
y : R
f : MvPolynomial (Fin (n + 1)) R
φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
let src := mapRingHom (eval s);
{
toRingHom :=
{ toMonoidHom := ↑src, map... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [aeval_X, Fin.cons_zero, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_comp,
Polynomial.coe_aeval_eq_eval, Polynomial.map_C, AlgHom.coe_mk, RingHom.toFun_eq_coe,
Polynomial.coe_mapRingHom, comp_apply, finSuccEquiv_apply, eval₂Hom_X',
Fin.cases_zero, Polynomial.map_X, Polynomial.eval_X, Fin.cons_succ,
F... | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{... | Mathlib.Data.MvPolynomial.Equiv.387_0.88gPfxLltQQTcHM | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
s' : Fin n → R
f : (MvPolynomial (Fin n) R)[X]
i : ℕ
⊢ Polynomial.coeff (Polynomial.map (eval s') f) i = (eval s') (Polynomial.coeff f i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [Polynomial.coeff_map] | theorem coeff_eval_eq_eval_coeff (s' : Fin n → R) (f : Polynomial (MvPolynomial (Fin n) R))
(i : ℕ) : Polynomial.coeff (Polynomial.map (eval s') f) i = eval s' (Polynomial.coeff f i) := by
| Mathlib.Data.MvPolynomial.Equiv.411_0.88gPfxLltQQTcHM | theorem coeff_eval_eq_eval_coeff (s' : Fin n → R) (f : Polynomial (MvPolynomial (Fin n) R))
(i : ℕ) : Polynomial.coeff (Polynomial.map (eval s') f) i = eval s' (Polynomial.coeff f i) | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
⊢ m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) ↔ cons i m ∈ support f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | apply Iff.intro | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support := by
| Mathlib.Data.MvPolynomial.Equiv.416_0.88gPfxLltQQTcHM | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support | Mathlib_Data_MvPolynomial_Equiv |
case mp
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
⊢ m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) → cons i m ∈ support f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro h | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support := by
apply Iff.intro
· | Mathlib.Data.MvPolynomial.Equiv.416_0.88gPfxLltQQTcHM | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support | Mathlib_Data_MvPolynomial_Equiv |
case mp
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
h : m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i)
⊢ cons i m ∈ support f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa [← finSuccEquiv_coeff_coeff] using h | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support := by
apply Iff.intro
· intro h
| Mathlib.Data.MvPolynomial.Equiv.416_0.88gPfxLltQQTcHM | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support | Mathlib_Data_MvPolynomial_Equiv |
case mpr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
⊢ cons i m ∈ support f → m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro h | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support := by
apply Iff.intro
· intro h
simpa [← finSuccEquiv_coeff_coeff] using h
· | Mathlib.Data.MvPolynomial.Equiv.416_0.88gPfxLltQQTcHM | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support | Mathlib_Data_MvPolynomial_Equiv |
case mpr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
h : cons i m ∈ support f
⊢ m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa [mem_support_iff, ← finSuccEquiv_coeff_coeff m f i] using h | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support := by
apply Iff.intro
· intro h
simpa [← finSuccEquiv_coeff_coeff] using h
· intro h
| Mathlib.Data.MvPolynomial.Equiv.416_0.88gPfxLltQQTcHM | theorem support_coeff_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} {m : Fin n →₀ ℕ} :
m ∈ (Polynomial.coeff ((finSuccEquiv R n) f) i).support ↔ Finsupp.cons i m ∈ f.support | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
⊢ totalDegree (Polynomial.coeff ((finSuccEquiv R n) f) i) + i ≤ totalDegree f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have hf'_sup : ((finSuccEquiv R n f).coeff i).support.Nonempty := by
rw [Finset.nonempty_iff_ne_empty, ne_eq, support_eq_empty]
exact hi | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
⊢ Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Finset.nonempty_iff_ne_empty, ne_eq, support_eq_empty] | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
⊢ ¬Polynomial.coeff ((finSuccEquiv R n) f) i = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact hi | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
⊢ totalDegree (Polynomial.coe... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have ⟨σ, hσ1, hσ2⟩ := Finset.exists_mem_eq_sup (support _) hf'_sup
(fun s => Finsupp.sum s fun _ e => e) | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ✝ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ✝ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
σ : Fin n →₀ ℕ
hσ1 : σ ∈ su... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | let σ' : Fin (n+1) →₀ ℕ := cons i σ | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ✝ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ✝ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
σ : Fin n →₀ ℕ
hσ1 : σ ∈ su... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | convert le_totalDegree (s := σ') _ | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
case h.e'_3
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ✝ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ✝ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
σ : Fin n →₀ ℕ
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [totalDegree, hσ2, sum_cons, add_comm] | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
case convert_4
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ✝ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ✝ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
σ : Fin n →₀... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← support_coeff_finSuccEquiv] | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
case convert_4
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ✝ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ✝ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
hi : Polynomial.coeff ((finSuccEquiv R n) f) i ≠ 0
hf'_sup : Finset.Nonempty (support (Polynomial.coeff ((finSuccEquiv R n) f) i))
σ : Fin n →₀... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact hσ1 | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib.Data.MvPolynomial.Equiv.425_0.88gPfxLltQQTcHM | /--
The `totalDegree` of a multivariable polynomial `p` is at least `i` more than the `totalDegree` of
the `i`th coefficient of `finSuccEquiv` applied to `p`, if this is nonzero.
-/
lemma totalDegree_coeff_finSuccEquiv_add_le (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ)
(hi : (finSuccEquiv R n f).coeff i ≠ 0) :
t... | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
⊢ Polynomial.support ((finSuccEquiv R n) f) = Finset.image (fun m => m 0) (support f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext i | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
| Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ i ∈ Polynomial.support ((finSuccEquiv R n) f) ↔ i ∈ Finset.image (fun m => m 0) (support f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff] | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
| Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ (∃ a, (Polynomial.coeff ((finSuccEquiv R n) f) i) a ≠ 0 a) ↔ ∃ a ∈ support f, a 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | constructor | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
| Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mp
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ (∃ a, (Polynomial.coeff ((finSuccEquiv R n) f) i) a ≠ 0 a) → ∃ a ∈ support f, a 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro ⟨m, hm⟩ | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· | Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mp.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
hm : (Polynomial.coeff ((finSuccEquiv R n) f) i) m ≠ 0 m
⊢ ∃ a ∈ support f, a 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine' ⟨cons i m, _, cons_zero _ _⟩ | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
| Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mp.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
hm : (Polynomial.coeff ((finSuccEquiv R n) f) i) m ≠ 0 m
⊢ cons i m ∈ support f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← support_coeff_finSuccEquiv] | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine' ⟨cons i m, _, cons_zero _ _⟩
| Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mp.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin n →₀ ℕ
hm : (Polynomial.coeff ((finSuccEquiv R n) f) i) m ≠ 0 m
⊢ m ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa using hm | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine' ⟨cons i m, _, cons_zero _ _⟩
rw... | Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mpr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ (∃ a ∈ support f, a 0 = i) → ∃ a, (Polynomial.coeff ((finSuccEquiv R n) f) i) a ≠ 0 a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro ⟨m, h, rfl⟩ | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine' ⟨cons i m, _, cons_zero _ _⟩
rw... | Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mpr.intro.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
m : Fin (n + 1) →₀ ℕ
h : m ∈ support f
⊢ ∃ a, (Polynomial.coeff ((finSuccEquiv R n) f) (m 0)) a ≠ 0 a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine' ⟨tail m, _⟩ | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine' ⟨cons i m, _, cons_zero _ _⟩
rw... | Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
case a.mpr.intro.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
m : Fin (n + 1) →₀ ℕ
h : m ∈ support f
⊢ (Polynomial.coeff ((finSuccEquiv R n) f) (m 0)) (tail m) ≠ 0 (tail m) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rwa [← coeff, zero_apply, ← mem_support_iff, support_coeff_finSuccEquiv, cons_tail] | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine' ⟨cons i m, _, cons_zero _ _⟩
rw... | Mathlib.Data.MvPolynomial.Equiv.445_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
⊢ Finset.image (cons i) (support (Polynomial.coeff ((finSuccEquiv R n) f) i)) =
Finset.filter (fun m => m 0 = i) (support f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext m | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
⊢ m ∈ Finset.image (cons i) (support (Polynomial.coeff ((finSuccEquiv R n) f) i)) ↔
m ∈ Finset.filter (fun m => m 0 = i) (suppor... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Finset.mem_filter, Finset.mem_image, mem_support_iff] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
⊢ (∃ a ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i), cons i a = m) ↔ coeff m f ≠ 0 ∧ m 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | conv_lhs =>
congr
ext
rw [mem_support_iff, finSuccEquiv_coeff_coeff, Ne.def] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
| ∃ a ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i), cons i a = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | congr
ext
rw [mem_support_iff, finSuccEquiv_coeff_coeff, Ne.def] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
| ∃ a ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i), cons i a = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | congr
ext
rw [mem_support_iff, finSuccEquiv_coeff_coeff, Ne.def] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
| ∃ a ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i), cons i a = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | congr | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case p
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
| fun a => a ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) ∧ cons i a = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | ext | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case p.h
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
x✝ : Fin n →₀ ℕ
| x✝ ∈ support (Polynomial.coeff ((finSuccEquiv R n) f) i) ∧ cons i x✝ = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [mem_support_iff, finSuccEquiv_coeff_coeff, Ne.def] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
| Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
⊢ (∃ x, ¬coeff (cons i x) f = 0 ∧ cons i x = m) ↔ coeff m f ≠ 0 ∧ m 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | constructor | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a.mp
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
⊢ (∃ x, ¬coeff (cons i x) f = 0 ∧ cons i x = m) → coeff m f ≠ 0 ∧ m 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rintro ⟨m', ⟨h, hm'⟩⟩ | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a.mp.intro.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
m' : Fin n →₀ ℕ
h : ¬coeff (cons i m') f = 0
hm' : cons i m' = m
⊢ coeff m f ≠ 0 ∧ m 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [← hm'] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a.mp.intro.intro
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
m' : Fin n →₀ ℕ
h : ¬coeff (cons i m') f = 0
hm' : cons i m' = m
⊢ coeff (cons i m') f ≠ 0 ∧ (cons i m') 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact ⟨h, by rw [cons_zero]⟩ | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
m' : Fin n →₀ ℕ
h : ¬coeff (cons i m') f = 0
hm' : cons i m' = m
⊢ (cons i m') 0 = i | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [cons_zero] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a.mpr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
⊢ coeff m f ≠ 0 ∧ m 0 = i → ∃ x, ¬coeff (cons i x) f = 0 ∧ cons i x = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro h | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case a.mpr
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
h : coeff m f ≠ 0 ∧ m 0 = i
⊢ ∃ x, ¬coeff (cons i x) f = 0 ∧ cons i x = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | use tail m | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case h
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
h : coeff m f ≠ 0 ∧ m 0 = i
⊢ ¬coeff (cons i (tail m)) f = 0 ∧ cons i (tail m) = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← h.2, cons_tail] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
case h
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
i : ℕ
m : Fin (n + 1) →₀ ℕ
h : coeff m f ≠ 0 ∧ m 0 = i
⊢ ¬coeff m f = 0 ∧ m = m | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp [h.1] | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i := by
ext m
rw [Finset.mem_filter, Finset.mem_image, mem_support_iff]
conv_lhs =>
congr
ext
rw [mem... | Mathlib.Data.MvPolynomial.Equiv.459_0.88gPfxLltQQTcHM | theorem finSuccEquiv_support' {f : MvPolynomial (Fin (n + 1)) R} {i : ℕ} :
Finset.image (Finsupp.cons i) (Polynomial.coeff ((finSuccEquiv R n) f) i).support =
f.support.filter fun m => m 0 = i | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
⊢ Finset.Nonempty (Polynomial.support ((finSuccEquiv R n) f)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty] | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
| Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
⊢ ¬(finSuccEquiv R n) f = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | refine fun c => h ?_ | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
| Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
c : (finSuccEquiv R n) f = 0
⊢ f = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | let ii := (finSuccEquiv R n).symm | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
refine fun c => h ?_
| Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
c : (finSuccEquiv R n) f = 0
ii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)
⊢ f = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | calc
f = ii (finSuccEquiv R n f) := by
simpa only [← AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm
_ = ii 0 := by rw [c]
_ = 0 := by simp | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
refine fun c => h ?_
let ii := (finSuccEquiv R n).symm
| Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
c : (finSuccEquiv R n) f = 0
ii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)
⊢ f = ii ((finSuccEquiv... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa only [← AlgEquiv.invFun_eq_symm] using ((finSuccEquiv R n).left_inv f).symm | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
refine fun c => h ?_
let ii := (finSuccEquiv R n).symm
calc
f = ii (finSuccEquiv R n f) := by
... | Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
c : (finSuccEquiv R n) f = 0
ii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)
⊢ ii ((finSuccEquiv R n... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [c] | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
refine fun c => h ?_
let ii := (finSuccEquiv R n).symm
calc
f = ii (finSuccEquiv R n f) := by
... | Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
c : (finSuccEquiv R n) f = 0
ii : (MvPolynomial (Fin n) R)[X] ≃ₐ[R] MvPolynomial (Fin (n + 1)) R := AlgEquiv.symm (finSuccEquiv R n)
⊢ ii 0 = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty := by
simp only [Finset.nonempty_iff_ne_empty, Ne, Polynomial.support_eq_empty]
refine fun c => h ?_
let ii := (finSuccEquiv R n).symm
calc
f = ii (finSuccEquiv R n f) := by
... | Mathlib.Data.MvPolynomial.Equiv.478_0.88gPfxLltQQTcHM | theorem support_finSuccEquiv_nonempty {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).support.Nonempty | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
⊢ degree ((finSuccEquiv R n) f) = ↑(degreeOf 0 f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
| Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α : Type ?u.1456857} {β : Type ?u.1456860} (f : α → β), (fun x => x) ∘ f = f
⊢ degree ((finSuccEquiv R n) f) = ↑(degreeOf 0 f) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rfl | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
| Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α : Type ?u.1456857} {β : Type ?u.1456860} (f : α → β), (fun x => x) ∘ f = f
h₁ : ∀ {α : Type ?u.1456922} {β : Type ?u.1456925} (f : α → β), (f ∘ fu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have h₂ : WithBot.some = Nat.cast := rfl | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rf... | Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α : Type ?u.1456857} {β : Type ?u.1456860} (f : α → β), (fun x => x) ∘ f = f
h₁ : ∀ {α : Type ?u.1456922} {β : Type ?u.1456925} (f : α → β), (f ∘ fu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by
rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h₀] | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rf... | Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α : Type ?u.1456857} {β : Type ?u.1456860} (f : α → β), (fun x => x) ∘ f = f
h₁ : ∀ {α : Type ?u.1456922} {β : Type ?u.1456925} (f : α → β), (f ∘ fu... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h₀] | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rf... | Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
h : f ≠ 0
h₀ : ∀ {α β : Type} (f : α → β), (fun x => x) ∘ f = f
h₁ : ∀ {α : Type ?u.1456922} {β : Type ?u.1456925} (f : α → β), (f ∘ fun x => x) = f
h₂ : WithBot.some... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.degree, ← h', ← h₂, Finset.coe_sup_of_nonempty (support_finSuccEquiv_nonempty h),
Finset.max_eq_sup_coe, h₁] | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by
-- TODO: these should be lemmas
have h₀ : ∀ {α β : Type _} (f : α → β), (fun x => x) ∘ f = f := fun f => rfl
have h₁ : ∀ {α β : Type _} (f : α → β), f ∘ (fun x => x) = f := fun f => rf... | Mathlib.Data.MvPolynomial.Equiv.490_0.88gPfxLltQQTcHM | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f ≠ 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | by_cases c : f = 0 | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
| Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
case pos
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : f = 0
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero] | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
by_cases c : f = 0
· | Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ natDegree ((finSuccEquiv R n) f) = degreeOf 0 f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )] | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
by_cases c : f = 0
· rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
· | Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ f ≠ 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simpa only [Ne.def] | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
by_cases c : f = 0
· rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
· rw [Polynomial.natDegree, degree_finSuccEquiv (by | Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ WithBot.unbot' 0 ↑(degreeOf 0 f) = degreeOf 0 f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | erw [WithBot.unbot'_coe] | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
by_cases c : f = 0
· rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
· rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]
| Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
case neg
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
f : MvPolynomial (Fin (n + 1)) R
c : ¬f = 0
⊢ ↑(degreeOf 0 f) = degreeOf 0 f | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | simp | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f := by
by_cases c : f = 0
· rw [c, (finSuccEquiv R n).map_zero, Polynomial.natDegree_zero, degreeOf_zero]
· rw [Polynomial.natDegree, degree_finSuccEquiv (by simpa only [Ne.def] )]
erw [WithBot... | Mathlib.Data.MvPolynomial.Equiv.503_0.88gPfxLltQQTcHM | theorem natDegree_finSuccEquiv (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).natDegree = degreeOf 0 f | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ degreeOf j (Polynomial.coeff ((finSuccEquiv R n) p) i) ≤ degreeOf (Fin.succ j) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff] | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
| Mathlib.Data.MvPolynomial.Equiv.512_0.88gPfxLltQQTcHM | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
⊢ ∀ b ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i), b j ≤ Finset.sup (support p) fun m => m (Fin.succ j) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | intro m hm | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
| Mathlib.Data.MvPolynomial.Equiv.512_0.88gPfxLltQQTcHM | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i)
⊢ m j ≤ Finset.sup (support p) fun m => m (Fin.succ j) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | rw [← Finsupp.cons_succ j i m] | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
intro m hm
| Mathlib.Data.MvPolynomial.Equiv.512_0.88gPfxLltQQTcHM | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p | Mathlib_Data_MvPolynomial_Equiv |
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
n : ℕ
p : MvPolynomial (Fin (n + 1)) R
j : Fin n
i : ℕ
m : Fin n →₀ ℕ
hm : m ∈ support (Polynomial.coeff ((finSuccEquiv R n) p) i)
⊢ (cons i m) (Fin.succ j) ≤ Finset.sup (support p) fun m => m (Fin.succ j) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Rename
import Mathlib.Data.Polynomial.AlgebraMap
import Mathlib.Data.MvPolynomial.Variables
import Ma... | exact Finset.le_sup
(f := fun (g : Fin (Nat.succ n) →₀ ℕ) => g (Fin.succ j))
(support_coeff_finSuccEquiv.1 hm) | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p := by
rw [degreeOf_eq_sup, degreeOf_eq_sup, Finset.sup_le_iff]
intro m hm
rw [← Finsupp.cons_succ j i m]
| Mathlib.Data.MvPolynomial.Equiv.512_0.88gPfxLltQQTcHM | theorem degreeOf_coeff_finSuccEquiv (p : MvPolynomial (Fin (n + 1)) R) (j : Fin n) (i : ℕ) :
degreeOf j (Polynomial.coeff (finSuccEquiv R n p) i) ≤ degreeOf j.succ p | Mathlib_Data_MvPolynomial_Equiv |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
F : Presheaf C X
U V : Opens ↑X
i : U ⟶ V
x : ↥U
inst✝ : ConcreteCategory C
s : (forget C).obj (F.obj (op V))
⊢ (germ F x) ((F.map i.op) s) = (germ F ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑V) }) x)) s | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [← comp_apply, germ_res] | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s := by | Mathlib.Topology.Sheaves.Stalks.114_0.hsVUPKIHRY0xmFk | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : (OpenNhds x)ᵒᵖ
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Op... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | induction' U using Opposite.rec with U | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
... | Mathlib.Topology.Sheaves.Stalks.119_0.hsVUPKIHRY0xmFk | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ | Mathlib_Topology_Sheaves_Stalks |
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : OpenNhds x
⊢ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | cases' U with U hxU | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
... | Mathlib.Topology.Sheaves.Stalks.119_0.hsVUPKIHRY0xmFk | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ | Mathlib_Topology_Sheaves_Stalks |
case mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y✝ Z : TopCat
F : Presheaf C X
x : ↑X
Y : C
f₁ f₂ : stalk F x ⟶ Y
ih :
∀ (U : Opens ↑X) (hxU : x ∈ U), germ F { val := x, property := hxU } ≫ f₁ = germ F { val := x, property := hxU } ≫ f₂
U : Opens ↑X
hxU : x ∈ U
⊢ colimit.ι (((whiskeringLeft (... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact ih U hxU | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=
... | Mathlib.Topology.Sheaves.Stalks.119_0.hsVUPKIHRY0xmFk | /-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
(ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x : ↑X
⊢ stalk (f _* F) (f x) ⟶ stalk F x | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | refine' ?_ ≫ colimit.pre _ (OpenNhds.map f x).op | /-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x := by
-- This is a hack; Lean doesn't like to elaborate the ter... | Mathlib.Topology.Sheaves.Stalks.139_0.hsVUPKIHRY0xmFk | /-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
x : ↑X
⊢ stalk (f _* F) (f x) ⟶
colimit
((OpenNhds.map f x).op ⋙ ((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) | /-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x := by
-- This is a hack; Lean doesn't like to elaborate the ter... | Mathlib.Topology.Sheaves.Stalks.139_0.hsVUPKIHRY0xmFk | /-- For a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the
stalk of `f _ * F` at `f x` and the stalk of `F` at `x`.
-/
def stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ germ (f _* F) { val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ≫ stalkPushforward C f F ↑x = germ F x | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whiskerRight_app] | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
(x : (Opens.map f).obj U) :
(f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
| Mathlib.Topology.Sheaves.Stalks.150_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ F.map
((NatTrans.op (OpenNhds.inclusionMapIso f ↑x).inv).app
(op { obj := U, property := (_ : ↑{ val := f ↑x, property := (_ : ↑x ∈ (Opens.map f).obj U) } ∈ U... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [CategoryTheory.Functor.map_id, Category.id_comp] | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
(x : (Opens.map f).obj U) :
(f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, wh... | Mathlib.Topology.Sheaves.Stalks.150_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C X
U : Opens ↑Y
x : ↥((Opens.map f).obj U)
⊢ colimit.ι (((whiskeringLeft (OpenNhds ↑x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ↑x).op).obj F)
((OpenNhds.map f ↑x).op.obj
(op { obj := U, property := (_ : ↑{... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rfl | @[reassoc (attr := simp), elementwise (attr := simp)]
theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
(x : (Opens.map f).obj U) :
(f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by
rw [stalkPushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, wh... | Mathlib.Topology.Sheaves.Stalks.150_0.hsVUPKIHRY0xmFk | @[reassoc (attr | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
| Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
⊢ stalkPushforward C (𝟙 X) ℱ x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext1 j | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
| Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
j : (OpenNhds ((𝟙 X) x))ᵒᵖ
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
j ≫
stalkPushforward C (𝟙 X) ℱ x =
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | induction' j with j | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext1 j
| Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
case w.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
j : OpenNhds ((𝟙 X) x)
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion ((𝟙 X) x)).op).obj (𝟙 X _* ℱ))
(op j) ≫
stalkPushforward C (𝟙 X) ℱ x ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rcases j with ⟨⟨_, _⟩, _⟩ | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext1 j
induction' j with j
| Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
case w.h.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
carrier✝ : Set ↑X
is_open'✝ : IsOpen carrier✝
property✝ : (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((𝟙 X) x))ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhd... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | erw [colimit.ι_map_assoc] | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext1 j
induction' j with j
rcases j with ⟨⟨_, _⟩, _⟩
| Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
case w.h.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
x : ↑X
carrier✝ : Set ↑X
is_open'✝ : IsOpen carrier✝
property✝ : (𝟙 X) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv) ℱ).app
(op {... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp [stalkFunctor, stalkPushforward] | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by
-- Porting note: We need to this to help ext tactic.
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext1 j
induction' j with j
rcases j with ⟨⟨_, _⟩, _⟩
erw [colimit.ι_ma... | Mathlib.Topology.Sheaves.Stalks.176_0.hsVUPKIHRY0xmFk | @[simp]
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _) | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
| Mathlib.Topology.Sheaves.Stalks.191_0.hsVUPKIHRY0xmFk | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) (f x) ≫ stalkPushforward C f ℱ x | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | ext U | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
| Mathlib.Topology.Sheaves.Stalks.191_0.hsVUPKIHRY0xmFk | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x | Mathlib_Topology_Sheaves_Stalks |
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
U : (OpenNhds ((f ≫ g) x))ᵒᵖ
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj (OpenNhds.inclusion ((f ≫ g) x)).op).obj
((f ≫ g) _* ℱ))
U ≫
... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | rcases U with ⟨⟨_, _⟩, _⟩ | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext U
| Mathlib.Topology.Sheaves.Stalks.191_0.hsVUPKIHRY0xmFk | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x | Mathlib_Topology_Sheaves_Stalks |
case w.mk.mk.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
ℱ : Presheaf C X
f : X ⟶ Y
g : Y ⟶ Z
x : ↑X
carrier✝ : Set ↑Z
is_open'✝ : IsOpen carrier✝
property✝ : (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((f ≫ g) x))ᵒᵖ (Open... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | simp [stalkFunctor, stalkPushforward] | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x := by
change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)
ext U
rcases U with ⟨⟨_, _⟩, _⟩
| Mathlib.Topology.Sheaves.Stalks.191_0.hsVUPKIHRY0xmFk | @[simp]
theorem comp (ℱ : X.Presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
ℱ.stalkPushforward C (f ≫ g) x =
(f _* ℱ).stalkPushforward C g (f x) ≫ ℱ.stalkPushforward C f x | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
⊢ IsIso (stalkPushforward C f F x) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
| Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
⊢ IsIso (stalkPushforward C f F x) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
((OpenNhds.inclusion (f x)).op ⋙ f _* F) :
_).symm ≪≫
colim.mapIso _) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
| Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case h.e'_5.h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
e_3✝ : stalk (f _* F) (f x) = colimit ((OpenNhds.inclusion (f x)).op ⋙ f _* F)
e_4✝ : stalk F x = colim.obj... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | swap | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
⊢ (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op ⋙ (OpenNhds.inclusion (f x)).op ⋙ f _* F ≅
((whisker... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | fapply NatIso.ofComponents | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.app
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
⊢ (X_1 : (OpenNhds x)ᵒᵖ) →
((IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op ⋙ (OpenNhds.inclusion ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | intro U | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.app
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
U : (OpenNhds x)ᵒᵖ
⊢ ((IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op ⋙ (OpenNhds.inclusion (f x)).op ... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | refine' F.mapIso (eqToIso _) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.app
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
U : (OpenNhds x)ᵒᵖ
⊢ (Opens.map f).op.obj ((OpenNhds.inclusion (f x)).op.obj ((IsOpenMap.functorNhds (... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | dsimp only [Functor.op] | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
case convert_2.app
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
hf : OpenEmbedding ⇑f
F : Presheaf C X
x : ↑X
this : Functor.Initial (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x)
U : (OpenNhds x)ᵒᵖ
⊢ op
((Opens.map f).obj
(op
((OpenNhds.inclusion (f x)).o... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | exact congr_arg op (Opens.ext <| Set.preimage_image_eq (unop U).1.1 hf.inj) | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) := by
haveI := Functor.initial_of_adjunction (hf.isOpenMap.adjunctionNhds x)
convert IsIso.of_iso
((Functor.Final.colimitIso (hf.isOpenMap.functorNhds x).op
... | Mathlib.Topology.Sheaves.Stalks.202_0.hsVUPKIHRY0xmFk | theorem stalkPushforward_iso_of_openEmbedding {f : X ⟶ Y} (hf : OpenEmbedding f) (F : X.Presheaf C)
(x : X) : IsIso (F.stalkPushforward _ f x) | Mathlib_Topology_Sheaves_Stalks |
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