Theorem: monom_coeff_eq_of_coord_eq (UnitConjecture/FreeModule.lean)

State before:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
⊢ coords [(a₀, x)] = coords [(a₁, x)] → a₀ = a₁
Tactic:
intro hyp
State after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
⊢ a₀ = a₁

State before:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
⊢ a₀ = a₁
Tactic:
let h₁ := congrFun hyp x
State after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
h₁ : coords [(a₀, x)] x = coords [(a₁, x)] x := congrFun hyp x
⊢ a₀ = a₁

State before:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
h₁ : coords [(a₀, x)] x = coords [(a₁, x)] x := congrFun hyp x
⊢ a₀ = a₁
Tactic:
simp [coords, monomCoeff] at h₁
State after:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
h₁ : a₀ = a₁
⊢ a₀ = a₁

State before:
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
x : X
a₀ a₁ : R
hyp : coords [(a₀, x)] = coords [(a₁, x)]
h₁ : a₀ = a₁
⊢ a₀ = a₁
Tactic:
assumption
State after:
no goals