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Mirrors > Home > MPE Home > Th. List > tmdmnd | Structured version Visualization version GIF version |
Description: A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
tmdmnd | ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
2 | eqid 2823 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
3 | 1, 2 | istmd 22684 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))) |
4 | 3 | simp1bi 1141 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 TopOpenctopn 16697 +𝑓cplusf 17851 Mndcmnd 17913 TopSpctps 21542 Cn ccn 21834 ×t ctx 22170 TopMndctmd 22680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-tmd 22682 |
This theorem is referenced by: tmdmulg 22702 tmdgsum 22705 oppgtmd 22707 prdstmdd 22734 tsmsxp 22765 xrge0iifmhm 31184 esumcst 31324 |
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