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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 24104 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
2 | tgptopon.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | 2, 3 | istps 22956 | . 2 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 1, 4 | sylib 218 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Basecbs 17245 TopOpenctopn 17468 TopOnctopon 22932 TopSpctps 22954 TopGrpctgp 24095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-top 22916 df-topon 22933 df-topsp 22955 df-tmd 24096 df-tgp 24097 |
This theorem is referenced by: tgpsubcn 24114 tgpmulg 24117 tgpmulg2 24118 subgtgp 24129 subgntr 24131 opnsubg 24132 clssubg 24133 clsnsg 24134 cldsubg 24135 tgpconncompeqg 24136 tgpconncomp 24137 tgpconncompss 24138 snclseqg 24140 tgphaus 24141 tgpt1 24142 tgpt0 24143 qustgpopn 24144 qustgplem 24145 qustgphaus 24147 prdstgpd 24149 tgptsmscld 24175 tsmsxplem1 24177 pl1cn 33916 |
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