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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 22977 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
2 | tgptopon.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | 2, 3 | istps 21831 | . 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 1, 4 | sylib 221 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 Basecbs 16760 TopOpenctopn 16926 TopOnctopon 21807 TopSpctps 21829 TopGrpctgp 22968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-top 21791 df-topon 21808 df-topsp 21830 df-tmd 22969 df-tgp 22970 |
This theorem is referenced by: tgpsubcn 22987 tgpmulg 22990 tgpmulg2 22991 subgtgp 23002 subgntr 23004 opnsubg 23005 clssubg 23006 clsnsg 23007 cldsubg 23008 tgpconncompeqg 23009 tgpconncomp 23010 tgpconncompss 23011 snclseqg 23013 tgphaus 23014 tgpt1 23015 tgpt0 23016 qustgpopn 23017 qustgplem 23018 qustgphaus 23020 prdstgpd 23022 tgptsmscld 23048 tsmsxplem1 23050 pl1cn 31619 |
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