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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 2818 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
2 | eqid 2818 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | istgp 22613 | . 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 6348 (class class class)co 7145 TopOpenctopn 16683 Grpcgrp 18041 invgcminusg 18042 Cn ccn 21760 TopMndctmd 22606 TopGrpctgp 22607 |
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 2790 ax-nul 5201 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-tgp 22609 |
This theorem is referenced by: grpinvhmeo 22622 istgp2 22627 oppgtgp 22634 tgplacthmeo 22639 subgtgp 22641 subgntr 22642 opnsubg 22643 clssubg 22644 cldsubg 22646 tgpconncompeqg 22647 tgpconncomp 22648 snclseqg 22651 tgphaus 22652 tgpt1 22653 tgpt0 22654 qustgpopn 22655 qustgplem 22656 qustgphaus 22658 prdstgpd 22660 tsmsinv 22683 tsmssub 22684 tgptsmscls 22685 tsmsxplem1 22688 tsmsxplem2 22689 |
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