Theorem: _private.Mathlib.Algebra.Module.GradedModule.0.DirectSum.Gmodule.one_smul'
File: Mathlib/Algebra/Module/GradedModule.lean

Proof steps:
state_before: ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
⊢ 1 • x = x
state_after: ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
⊢ (smulAddMonoidHom A M) 1 = AddMonoidHom.id (⨁ (i : ιB), M i)
tactic: suffices smulAddMonoidHom A M 1 = AddMonoidHom.id (⨁ i, M i) from DFunLike.congr_fun this x
state_before: ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
⊢ (smulAddMonoidHom A M) 1 = AddMonoidHom.id (⨁ (i : ιB), M i)
state_after: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
⊢ ∀ (i : ιB) (y : M i), ((smulAddMonoidHom A M) 1) ((of M i) y) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) y)
tactic: apply DirectSum.addHom_ext
state_before: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
⊢ ∀ (i : ιB) (y : M i), ((smulAddMonoidHom A M) 1) ((of M i) y) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) y)
state_after: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ ((smulAddMonoidHom A M) 1) ((of M i) xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
tactic: intro i xi
state_before: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ ((smulAddMonoidHom A M) 1) ((of M i) xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
state_after: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ ((smulAddMonoidHom A M) ((of A 0) GOne.one)) ((of M i) xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
tactic: rw [show (1 : DirectSum ιA fun i => A i) = (of A 0) GOne.one by rfl]
state_before: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ ((smulAddMonoidHom A M) ((of A 0) GOne.one)) ((of M i) xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
state_after: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ (of M (0 +ᵥ i)) (GSMul.smul GOne.one xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
tactic: rw [smulAddMonoidHom_apply_of_of]
state_before: case H
ιA : Type u_1
ιB : Type u_2
A : ιA → Type u_3
M : ιB → Type u_4
inst✝⁷ : AddMonoid ιA
inst✝⁶ : VAdd ιA ιB
inst✝⁵ : (i : ιA) → AddCommMonoid (A i)
inst✝⁴ : (i : ιB) → AddCommMonoid (M i)
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιB
inst✝¹ : GMonoid A
inst✝ : Gmodule A M
x : ⨁ (i : ιB), M i
i : ιB
xi : M i
⊢ (of M (0 +ᵥ i)) (GSMul.smul GOne.one xi) = (AddMonoidHom.id (⨁ (i : ιB), M i)) ((of M i) xi)
state_after: no goals
tactic: exact DirectSum.of_eq_of_gradedMonoid_eq (one_smul (GradedMonoid A) <| GradedMonoid.mk i xi)