-- Mathlib/RingTheory/Smooth/Kaehler.lean
lemma isScalarTower_of_section_of_ker_sqZero :
    letI := g.toRingHom.toAlgebra; IsScalarTower P S (RingHom.ker (algebraMap P S))

Proof state:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) * ↑m = p * ↑m

Tactic:
rw [← sub_eq_zero, ← Ideal.mem_bot, ← hf', pow_two, ← sub_mul]

Result:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ (g ((algebraMap P S) p) - p) * ↑m ∈ RingHom.ker (algebraMap P S) * RingHom.ker (algebraMap P S)

Proof state:
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
⊢ IsScalarTower P S ↥(RingHom.ker (algebraMap P S))

Tactic:
constructor

Result:
case smul_assoc
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
⊢ ∀ (x : P) (y : S) (z : ↥(RingHom.ker (algebraMap P S))), (x • y) • z = x • y • z

Proof state:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g s * (g ((algebraMap P S) p) * ↑m) = g s * (p * ↑m)

Tactic:
congr 1

Result:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) * ↑m = p * ↑m

Proof state:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) - p ∈ RingHom.ker (algebraMap P S)

Tactic:
simpa [RingHom.mem_ker, sub_eq_zero] using AlgHom.congr_fun hg (algebraMap P S p)

Result:
no goals

Proof state:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g (p • s) * ↑m = p * (g s * ↑m)

Tactic:
simp only [Algebra.smul_def, map_mul, mul_assoc, mul_left_comm _ (g s)]

Result:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g s * (g ((algebraMap P S) p) * ↑m) = g s * (p * ↑m)

Proof state:
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
⊢ IsScalarTower P S ↥(RingHom.ker (algebraMap P S))

Tactic:
letI := g.toRingHom.toAlgebra

Result:
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
⊢ IsScalarTower P S ↥(RingHom.ker (algebraMap P S))

Proof state:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ ↑((p • s) • m) = ↑(p • s • m)

Tactic:
show g (p • s) * m = p * (g s * m)

Result:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g (p • s) * ↑m = p * (g s * ↑m)

Proof state:
case smul_assoc
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ (p • s) • m = p • s • m

Tactic:
ext

Result:
case smul_assoc.a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ ↑((p • s) • m) = ↑(p • s • m)

Proof state:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ (g ((algebraMap P S) p) - p) * ↑m ∈ RingHom.ker (algebraMap P S) * RingHom.ker (algebraMap P S)

Tactic:
refine Ideal.mul_mem_mul ?_ m.2

Result:
case smul_assoc.a.e_a
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) - p ∈ RingHom.ker (algebraMap P S)