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Mirrors > Home > MPE Home > Th. List > tmdcn | Structured version Visualization version GIF version |
Description: In a topological monoid, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpcn.1 | ⊢ 𝐹 = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
tmdcn | ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpcn.1 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
2 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | 1, 2 | istmd 22679 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
4 | 3 | simp3bi 1144 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 TopOpenctopn 16687 +𝑓cplusf 17841 Mndcmnd 17903 TopSpctps 21537 Cn ccn 21829 ×t ctx 22165 TopMndctmd 22675 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-tmd 22677 |
This theorem is referenced by: tgpcn 22689 cnmpt1plusg 22692 cnmpt2plusg 22693 tmdcn2 22694 submtmd 22709 tsmsadd 22752 mulrcn 22784 mhmhmeotmd 31280 xrge0pluscn 31293 |
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