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 ) )