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Mirrors > Home > MPE Home > Th. List > tgptps | Structured version Visualization version GIF version |
Description: A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgptps | ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptmd 22690 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tmdtps 22687 | . 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 TopSpctps 21543 TopMndctmd 22681 TopGrpctgp 22682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-tmd 22683 df-tgp 22684 |
This theorem is referenced by: tgptopon 22693 istgp2 22702 tsmsinv 22759 tsmssub 22760 tgptsmscls 22761 tgptsmscld 22762 tsmsxplem1 22764 tsmsxp 22766 trgtps 22781 nrgtrg 23302 |
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