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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 2818 | . . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
2 | eqid 2818 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
3 | 1, 2 | istmd 22610 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp2bi 1138 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 TopOpenctopn 16683 +𝑓cplusf 17837 Mndcmnd 17899 TopSpctps 21468 Cn ccn 21760 ×t ctx 22096 TopMndctmd 22606 |
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 2790 ax-nul 5201 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-tmd 22608 |
This theorem is referenced by: tgptps 22616 tmdtopon 22617 submtmd 22640 prdstmdd 22659 tsmsadd 22682 tsmssplit 22687 tlmtps 22723 |
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