Metamath Proof Explorer

Theorem tgptmd

Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015)

Ref Expression
Assertion tgptmd ⊢ G ∈ TopGrp → G ∈ TopMnd

Proof

Step Hyp Ref Expression
1 eqid ⊢ TopOpen ⁡ G = TopOpen ⁡ G
2 eqid ⊢ inv g ⁡ G = inv g ⁡ G
3 1 2 istgp ⊢ G ∈ TopGrp ↔ G ∈ Grp ∧ G ∈ TopMnd ∧ inv g ⁡ G ∈ TopOpen ⁡ G Cn TopOpen ⁡ G
4 3 simp2bi ⊢ G ∈ TopGrp → G ∈ TopMnd