subgraphOfAdj_connected — Mathlib

English

For any adjacency hvw in G, the subgraph of G induced by hvw is connected.

Русский

Для любой пары смежных вершин v,w в G, подграф, порождаемый gcd hvw, связан.

LaTeX

$$$\forall v,w : V\, (hvw : G.Adj v w)\;:\ (G.subgraphOfAdj hvw).Connected$$$

Lean4

@[simp]
theorem subgraphOfAdj_connected {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).Connected :=
  by
  refine ⟨⟨?_⟩⟩
  rintro ⟨a, ha⟩ ⟨b, hb⟩
  simp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb
  obtain rfl | rfl := ha <;> obtain rfl | rfl := hb <;>
    first
    | rfl
    | (apply Adj.reachable; simp)

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