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Mirrors > Home > MPE Home > Th. List > tmdtps | Structured version Visualization version GIF version |
Description: A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
tmdtps | ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
2 | eqid 2737 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
3 | 1, 2 | istmd 23377 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp2bi 1146 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 TopOpenctopn 17263 +𝑓cplusf 18454 Mndcmnd 18516 TopSpctps 22233 Cn ccn 22527 ×t ctx 22863 TopMndctmd 23373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7354 df-tmd 23375 |
This theorem is referenced by: tgptps 23383 tmdtopon 23384 submtmd 23407 prdstmdd 23427 tsmsadd 23450 tsmssplit 23455 tlmtps 23491 |
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