tgptopon - Metamath Proof Explorer

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Theorem tgptopon 24309

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

**Hypotheses**
Ref	Expression
tgpcn.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgptopon.x	⊢ 𝑋 = (Base‘𝐺)

**Assertion**
Ref	Expression
tgptopon	⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))

**Proof of Theorem tgptopon**
Step	Hyp	Ref	Expression
1		tgptps 24307	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2		tgptopon.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
4	2, 3	istps 23160	. 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	sylib 221	1 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ‘cfv 6533 Basecbs 17302 TopOpenctopn 17507 TopOnctopon 23136 TopSpctps 23158 TopGrpctgp 24298

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-ov 7417 df-top 23120 df-topon 23137 df-topsp 23159 df-tmd 24299 df-tgp 24300

This theorem is used by: tgpsubcn 24317 tgpmulg 24320 tgpmulg2 24321 subgtgp 24332 subgntr 24334 opnsubg 24335 clssubg 24336 clsnsg 24337 cldsubg 24338 tgpconncompeqg 24339 tgpconncomp 24340 tgpconncompss 24341 snclseqg 24343 tgphaus 24344 tgpt1 24345 tgpt0 24346 qustgpopn 24347 qustgplem 24348 qustgphaus 24350 prdstgpd 24352 tgptsmscld 24378 tsmsxplem1 24380 pl1cn 34466

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