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Mirrors > Home > MPE Home > Th. List > tgpgrp | Structured version Visualization version GIF version |
Description: A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgpgrp | ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
2 | eqid 2825 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | istgp 22251 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp1bi 1181 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 TopOpenctopn 16435 Grpcgrp 17776 invgcminusg 17777 Cn ccn 21399 TopMndctmd 22244 TopGrpctgp 22245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-tgp 22247 |
This theorem is referenced by: grpinvhmeo 22260 istgp2 22265 oppgtgp 22272 tgplacthmeo 22277 subgtgp 22279 subgntr 22280 opnsubg 22281 clssubg 22282 cldsubg 22284 tgpconncompeqg 22285 tgpconncomp 22286 snclseqg 22289 tgphaus 22290 tgpt1 22291 tgpt0 22292 qustgpopn 22293 qustgplem 22294 qustgphaus 22296 prdstgpd 22298 tsmsinv 22321 tsmssub 22322 tgptsmscls 22323 tsmsxplem1 22326 tsmsxplem2 22327 |
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