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Mirrors > Home > MPE Home > Th. List > tgptopon | Structured version Visualization version GIF version |
Description: The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgptopon.x | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
tgptopon | ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptps 23553 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
2 | tgptopon.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | 2, 3 | istps 22405 | . 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 1, 4 | sylib 217 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 Basecbs 17131 TopOpenctopn 17354 TopOnctopon 22381 TopSpctps 22403 TopGrpctgp 23544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-top 22365 df-topon 22382 df-topsp 22404 df-tmd 23545 df-tgp 23546 |
This theorem is referenced by: tgpsubcn 23563 tgpmulg 23566 tgpmulg2 23567 subgtgp 23578 subgntr 23580 opnsubg 23581 clssubg 23582 clsnsg 23583 cldsubg 23584 tgpconncompeqg 23585 tgpconncomp 23586 tgpconncompss 23587 snclseqg 23589 tgphaus 23590 tgpt1 23591 tgpt0 23592 qustgpopn 23593 qustgplem 23594 qustgphaus 23596 prdstgpd 23598 tgptsmscld 23624 tsmsxplem1 23626 pl1cn 32866 |
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