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Metamath Proof Explorer

Theorem tgptopon

Description: The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014) (Revised by Mario Carneiro, 13-Aug-2015)

Ref Expression
Hypotheses tgpcn.j ⊢ J = TopOpen ⁡ G
tgptopon.x ⊢ X = Base G
Assertion tgptopon ⊢ G ∈ TopGrp → J ∈ TopOn ⁡ X

Proof

Step Hyp Ref Expression
1 tgpcn.j ⊢ J = TopOpen ⁡ G
2 tgptopon.x ⊢ X = Base G
3 tgptps ⊢ G ∈ TopGrp → G ∈ TopSp
4 2 1 istps ⊢ G ∈ TopSp ↔ J ∈ TopOn ⁡ X
5 3 4 sylib ⊢ G ∈ TopGrp → J ∈ TopOn ⁡ X