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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 2821 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
2 | eqid 2821 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | istgp 22615 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp1bi 1137 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 TopOpenctopn 16685 Grpcgrp 18043 invgcminusg 18044 Cn ccn 21762 TopMndctmd 22608 TopGrpctgp 22609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7148 df-tgp 22611 |
This theorem is referenced by: grpinvhmeo 22624 istgp2 22629 oppgtgp 22636 tgplacthmeo 22641 subgtgp 22643 subgntr 22644 opnsubg 22645 clssubg 22646 cldsubg 22648 tgpconncompeqg 22649 tgpconncomp 22650 snclseqg 22653 tgphaus 22654 tgpt1 22655 tgpt0 22656 qustgpopn 22657 qustgplem 22658 qustgphaus 22660 prdstgpd 22662 tsmsinv 22685 tsmssub 22686 tgptsmscls 22687 tsmsxplem1 22690 tsmsxplem2 22691 |
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