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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 24087 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
| 2 | tmdtps 24084 | . 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 2108 TopSpctps 22938 TopMndctmd 24078 TopGrpctgp 24079 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tmd 24080 df-tgp 24081 |
| This theorem is referenced by: tgptopon 24090 istgp2 24099 tsmsinv 24156 tsmssub 24157 tgptsmscls 24158 tgptsmscld 24159 tsmsxplem1 24161 tsmsxp 24163 trgtps 24178 nrgtrg 24711 |
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