Theorem: Algebra.Generators.comp_vars
File: Mathlib/RingTheory/Generators.lean

Proof steps:
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : Generators R S
T : Type ?u.37168
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
s : T
⊢ (aeval (Sum.elim Q.val (⇑(algebraMap S T) ∘ P.val)))
      ((fun x => Finsupp.sum (Q.σ x) fun n r => (rename Sum.inr) (P.σ r) * (monomial (Finsupp.mapDomain Sum.inl n)) 1)
        s) =
    s
state_after: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : Generators R S
T : Type ?u.37168
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
s : T
this : ∀ (x : P.Ring), (aeval (⇑(algebraMap S T) ∘ P.val)) x = (algebraMap S T) ((aeval P.val) x)
⊢ (aeval (Sum.elim Q.val (⇑(algebraMap S T) ∘ P.val)))
      ((fun x => Finsupp.sum (Q.σ x) fun n r => (rename Sum.inr) (P.σ r) * (monomial (Finsupp.mapDomain Sum.inl n)) 1)
        s) =
    s
tactic: have (x : P.Ring) : aeval (algebraMap S T ∘ P.val) x = algebraMap S T (aeval P.val x) := by
      rw [map_aeval, aeval_def, coe_eval₂Hom, ← IsScalarTower.algebraMap_eq, Function.comp]
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : Generators R S
T : Type ?u.37168
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
s : T
this : ∀ (x : P.Ring), (aeval (⇑(algebraMap S T) ∘ P.val)) x = (algebraMap S T) ((aeval P.val) x)
⊢ (aeval (Sum.elim Q.val (⇑(algebraMap S T) ∘ P.val)))
      ((fun x => Finsupp.sum (Q.σ x) fun n r => (rename Sum.inr) (P.σ r) * (monomial (Finsupp.mapDomain Sum.inl n)) 1)
        s) =
    s
state_after: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : Generators R S
T : Type ?u.37168
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
s : T
this : ∀ (x : P.Ring), (aeval (⇑(algebraMap S T) ∘ P.val)) x = (algebraMap S T) ((aeval P.val) x)
⊢ (aeval (Sum.elim Q.val (⇑(algebraMap S T) ∘ P.val)))
      ((fun x => Finsupp.sum (Q.σ x) fun n r => (rename Sum.inr) (P.σ r) * (monomial (Finsupp.mapDomain Sum.inl n)) 1)
        s) =
    (aeval Q.val) (Finsupp.sum (Q.σ s) AddMonoidAlgebra.single)
tactic: conv_rhs => rw [← Q.aeval_val_σ s, ← (Q.σ s).sum_single]
state_before: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : Generators R S
T : Type ?u.37168
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
s : T
this : ∀ (x : P.Ring), (aeval (⇑(algebraMap S T) ∘ P.val)) x = (algebraMap S T) ((aeval P.val) x)
⊢ (aeval (Sum.elim Q.val (⇑(algebraMap S T) ∘ P.val)))
      ((fun x => Finsupp.sum (Q.σ x) fun n r => (rename Sum.inr) (P.σ r) * (monomial (Finsupp.mapDomain Sum.inl n)) 1)
        s) =
    (aeval Q.val) (Finsupp.sum (Q.σ s) AddMonoidAlgebra.single)
state_after: no goals
tactic: simp only [map_finsupp_sum, _root_.map_mul, aeval_rename, Sum.elim_comp_inr, this, aeval_val_σ,
      aeval_monomial, _root_.map_one, Finsupp.prod_mapDomain_index_inj Sum.inl_injective,
      Sum.elim_inl, one_mul, single_eq_monomial]