tgptopon - Metamath Proof Explorer

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Theorem tgptopon 24456

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 24454	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2		tgptopon.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
4	2, 3	istps 23307	. 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 2147 ‘cfv 6566 Basecbs 17431 TopOpenctopn 17642 TopOnctopon 23283 TopSpctps 23305 TopGrpctgp 24445

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-iota 6520 df-fun 6568 df-fv 6574 df-ov 7454 df-top 23267 df-topon 23284 df-topsp 23306 df-tmd 24446 df-tgp 24447

This theorem is used by: tgpsubcn 24464 tgpmulg 24467 tgpmulg2 24468 subgtgp 24479 subgntr 24481 opnsubg 24482 clssubg 24483 clsnsg 24484 cldsubg 24485 tgpconncompeqg 24486 tgpconncomp 24487 tgpconncompss 24488 snclseqg 24490 tgphaus 24491 tgpt1 24492 tgpt0 24493 qustgpopn 24494 qustgplem 24495 qustgphaus 24497 prdstgpd 24499 tgptsmscld 24525 tsmsxplem1 24527 pl1cn 34698

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