Module: Mathlib.RingTheory.PolynomialLaw.Basic

Goal state before:
pp: case e_f
R : Type u
inst✝⁶ : CommSemiring R
M : Type u_1
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommMonoid N
inst✝² : Module R N
S : Type v
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
f : M →ₚₗ[R] N
s : Finset S
p : MvPolynomial (Fin s.card) R ⊗[R] M
s' : Finset S
p' : MvPolynomial (Fin s'.card) R ⊗[R] M
h✝ : π R M S ⟨s, p⟩ = π R M S ⟨s', p'⟩
u : ↥(φ R s).range ⊗[R] M := (rTensor M (φ R s).rangeRestrict.toLinearMap) p
hu : u = (rTensor M (φ R s).rangeRestrict.toLinearMap) p
uFG : (φ R s).range.FG
u' : ↥(φ R s').range ⊗[R] M := (rTensor M (φ R s').rangeRestrict.toLinearMap) p'
hu' : u' = (rTensor M (φ R s').rangeRestrict.toLinearMap) p'
u'FG : (φ R s').range.FG
huu' : (rTensor M (φ R s).range.val.toLinearMap) u = (rTensor M (φ R s').range.val.toLinearMap) u'
B : Subalgebra R S
hAB : (φ R s).range ≤ B
hA'B : (φ R s').range ≤ B
h : (rTensor M (Subalgebra.inclusion hAB).toLinearMap) u = (rTensor M (Subalgebra.inclusion hA'B).toLinearMap) u'
t : Finset S
hB : B = (φ R t).range
hAB' : (φ R s).range ≤ (φ R t).range
hA'B' : (φ R s').range ≤ (φ R t).range
q : MvPolynomial (Fin t.card) R ⊗[R] M
hq :
  (rTensor M (φ R t).rangeRestrict.toLinearMap) q =
    (rTensor M ((Subalgebra.inclusion ⋯).comp (Subalgebra.inclusion hAB)).toLinearMap) u
x✝¹ : ↥(φ R s).range
x✝ : M
⊢ ((AlgebraTensorModule.curry (rTensor M (Subalgebra.inclusion hAB').toLinearMap)) x✝¹) x✝ =
    ((AlgebraTensorModule.curry
          (rTensor M (Subalgebra.inclusion ⋯).toLinearMap ∘ₗ rTensor M (Subalgebra.inclusion hAB).toLinearMap))
        x✝¹)
      x✝

Tactic:
;

Goal state after:
case e_f
R : Type u
inst✝⁶ : CommSemiring R
M : Type u_1
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
N : Type u_2
inst✝³ : AddCommMonoid N
inst✝² : Module R N
S : Type v
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
f : M →ₚₗ[R] N
s : Finset S
p : MvPolynomial (Fin s.card) R ⊗[R] M
s' : Finset S
p' : MvPolynomial (Fin s'.card) R ⊗[R] M
h✝ : π R M S ⟨s, p⟩ = π R M S ⟨s', p'⟩
u : ↥(φ R s).range ⊗[R] M := (rTensor M (φ R s).rangeRestrict.toLinearMap) p
hu : u = (rTensor M (φ R s).rangeRestrict.toLinearMap) p
uFG : (φ R s).range.FG
u' : ↥(φ R s').range ⊗[R] M := (rTensor M (φ R s').rangeRestrict.toLinearMap) p'
hu' : u' = (rTensor M (φ R s').rangeRestrict.toLinearMap) p'
u'FG : (φ R s').range.FG
huu' : (rTensor M (φ R s).range.val.toLinearMap) u = (rTensor M (φ R s').range.val.toLinearMap) u'
B : Subalgebra R S
hAB : (φ R s).range ≤ B
hA'B : (φ R s').range ≤ B
h : (rTensor M (Subalgebra.inclusion hAB).toLinearMap) u = (rTensor M (Subalgebra.inclusion hA'B).toLinearMap) u'
t : Finset S
hB : B = (φ R t).range
hAB' : (φ R s).range ≤ (φ R t).range
hA'B' : (φ R s').range ≤ (φ R t).range
q : MvPolynomial (Fin t.card) R ⊗[R] M
hq :
  (rTensor M (φ R t).rangeRestrict.toLinearMap) q =
    (rTensor M ((Subalgebra.inclusion ⋯).comp (Subalgebra.inclusion hAB)).toLinearMap) u
x✝¹ : ↥(φ R s).range
x✝ : M
⊢ ((AlgebraTensorModule.curry (rTensor M (Subalgebra.inclusion hAB').toLinearMap)) x✝¹) x✝ =
    ((AlgebraTensorModule.curry
          (rTensor M (Subalgebra.inclusion ⋯).toLinearMap ∘ₗ rTensor M (Subalgebra.inclusion hAB).toLinearMap))
        x✝¹)
      x✝