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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 22617 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tmdtps 22614 | . 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 2105 TopSpctps 21470 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-tmd 22610 df-tgp 22611 |
This theorem is referenced by: tgptopon 22620 istgp2 22629 tsmsinv 22685 tsmssub 22686 tgptsmscls 22687 tgptsmscld 22688 tsmsxplem1 22690 tsmsxp 22692 trgtps 22707 nrgtrg 23228 |
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