Theorem: MvPowerSeries.map_algebraMap_eq_subst_X (DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean)

State before:
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
⊢ (map σ (algebraMap R S)) f = subst X f
Tactic:
ext e
State after:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (coeff S e) ((map σ (algebraMap R S)) f) = (coeff S e) (subst X f)

State before:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (coeff S e) ((map σ (algebraMap R S)) f) = (coeff S e) (subst X f)
Tactic:
rw [coeff_map, coeff_subst substDomain_X f e, finsum_eq_single _ e]
State after:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff R e) f) = (coeff R e) f • (coeff S e) (e.prod fun s e => X s ^ e)

case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff R x) f • (coeff S e) (x.prod fun s e => X s ^ e) = 0

State before:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ (algebraMap R S) ((coeff R e) f) = (coeff R e) f • (coeff S e) (e.prod fun s e => X s ^ e)
Tactic:
rw [← MvPowerSeries.monomial_one_eq, coeff_monomial_same,
  algebra_compatible_smul S, smul_eq_mul, mul_one]
State after:
no goals

State before:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e : σ →₀ ℕ
⊢ ∀ (x : σ →₀ ℕ), x ≠ e → (coeff R x) f • (coeff S e) (x.prod fun s e => X s ^ e) = 0
Tactic:
intro d hd
State after:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e d : σ →₀ ℕ
hd : d ≠ e
⊢ (coeff R d) f • (coeff S e) (d.prod fun s e => X s ^ e) = 0

State before:
case h
σ : Type u_1
inst✝⁸ : DecidableEq σ
A : Type u_2
inst✝⁷ : CommSemiring A
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra A R
τ : Type u_4
inst✝⁴ : DecidableEq τ
S : Type u_5
inst✝³ : CommRing S
inst✝² : Algebra A S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower A R S
a : σ → MvPowerSeries τ S
ha : SubstDomain a
f : MvPowerSeries σ R
e d : σ →₀ ℕ
hd : d ≠ e
⊢ (coeff R d) f • (coeff S e) (d.prod fun s e => X s ^ e) = 0
Tactic:
rw [← MvPowerSeries.monomial_one_eq, coeff_monomial_ne hd.symm, smul_zero]
State after:
no goals