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Mirrors > Home > MPE Home > Th. List > tgptmd | Structured version Visualization version GIF version |
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
tgptmd | ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
2 | eqid 2733 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | istgp 23573 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp2bi 1147 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7406 TopOpenctopn 17364 Grpcgrp 18816 invgcminusg 18817 Cn ccn 22720 TopMndctmd 23566 TopGrpctgp 23567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6493 df-fv 6549 df-ov 7409 df-tgp 23569 |
This theorem is referenced by: tgptps 23576 tgpcn 23580 tgpsubcn 23586 tgpmulg 23589 oppgtgp 23594 tgplacthmeo 23599 subgtgp 23601 clsnsg 23606 tgpt0 23615 prdstgpd 23621 tsmssub 23645 tsmsxp 23651 trgtmd2 23665 nlmtlm 24203 qqhcn 32960 |
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