case h.mk
α : Type u_1
β : Type u_2
γ : Type ?u.9830
G : SimpleGraph α
H : SimpleGraph β
x : α × β
a' : α
b' : β
⊢ (a', b') ∈ neighborSet (G □ H) x ↔ (a', b') ∈ neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd
α : Type u_1
β : Type u_2
γ : Type ?u.9830
G : SimpleGraph α
H : SimpleGraph β
x : α × β
⊢ neighborSet (G □ H) x = neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd
ext ⟨a', b'⟩
case h.mk
α : Type u_1
β : Type u_2
γ : Type ?u.9830
G : SimpleGraph α
H : SimpleGraph β
x : α × β
a' : α
b' : β
⊢ Adj G x.fst a' ∧ x.snd = b' ∨ Adj H x.snd b' ∧ x.fst = a' ↔ Adj G x.fst a' ∧ b' = x.snd ∨ a' = x.fst ∧ Adj H x.snd b'
case h.mk
α : Type u_1
β : Type u_2
γ : Type ?u.9830
G : SimpleGraph α
H : SimpleGraph β
x : α × β
a' : α
b' : β
⊢ (a', b') ∈ neighborSet (G □ H) x ↔ (a', b') ∈ neighborSet G x.fst ×ˢ {x.snd} ∪ {x.fst} ×ˢ neighborSet H x.snd
simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
no goals
case h.mk
α : Type u_1
β : Type u_2
γ : Type ?u.9830
G : SimpleGraph α
H : SimpleGraph β
x : α × β
a' : α
b' : β
⊢ Adj G x.fst a' ∧ x.snd = b' ∨ Adj H x.snd b' ∧ x.fst = a' ↔ Adj G x.fst a' ∧ b' = x.snd ∨ a' = x.fst ∧ Adj H x.snd b'
simp only [eq_comm, and_comm]