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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 21693 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tmdtps 21690 | . 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 1977 TopSpctps 20519 TopMndctmd 21684 TopGrpctgp 21685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-tmd 21686 df-tgp 21687 |
This theorem is referenced by: tgptopon 21696 istgp2 21705 tsmsinv 21761 tsmssub 21762 tgptsmscls 21763 tgptsmscld 21764 tsmsxplem1 21766 tsmsxp 21768 trgtps 21783 nrgtrg 22304 |
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