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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 2739 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
2 | eqid 2739 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | istgp 23005 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp1bi 1147 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ‘cfv 6400 (class class class)co 7234 TopOpenctopn 16958 Grpcgrp 18397 invgcminusg 18398 Cn ccn 22152 TopMndctmd 22998 TopGrpctgp 22999 |
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 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-nul 5215 |
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 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-iota 6358 df-fv 6408 df-ov 7237 df-tgp 23001 |
This theorem is referenced by: grpinvhmeo 23014 istgp2 23019 oppgtgp 23026 tgplacthmeo 23031 subgtgp 23033 subgntr 23035 opnsubg 23036 clssubg 23037 cldsubg 23039 tgpconncompeqg 23040 tgpconncomp 23041 snclseqg 23044 tgphaus 23045 tgpt1 23046 tgpt0 23047 qustgpopn 23048 qustgplem 23049 qustgphaus 23051 prdstgpd 23053 tsmsinv 23076 tsmssub 23077 tgptsmscls 23078 tsmsxplem1 23081 tsmsxplem2 23082 |
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