Theorem: LO.FirstOrder.Arith.Model.IsMapping.insert (Arithmetization/ISigmaOne/HFS/Basic.lean)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y m : M
h : IsMapping m
disjoint : x ∉ domain m
⊢ IsMapping (Insert.insert ⟪x, y⟫ m)
Tactic:
simp [insert_eq_union_singleton]
State after:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y m : M
h : IsMapping m
disjoint : x ∉ domain m
⊢ IsMapping ({⟪x, y⟫} ∪ m)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y m : M
h : IsMapping m
disjoint : x ∉ domain m
⊢ IsMapping ({⟪x, y⟫} ∪ m)
Tactic:
exact IsMapping.union_of_disjoint_domain (by simp) h (by simpa)
State after:
no goals

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y m : M
h : IsMapping m
disjoint : x ∉ domain m
⊢ IsMapping {⟪x, y⟫}
Tactic:
simp
State after:
no goals

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
x y m : M
h : IsMapping m
disjoint : x ∉ domain m
⊢ Disjoint (domain {⟪x, y⟫}) (domain m)
Tactic:
simpa
State after:
no goals