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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 22682 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
4 | 3 | simp3bi 1143 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 TopOpenctopn 16695 +𝑓cplusf 17849 Mndcmnd 17911 TopSpctps 21540 Cn ccn 21832 ×t ctx 22168 TopMndctmd 22678 |
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 2793 ax-nul 5210 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-tmd 22680 |
This theorem is referenced by: tgpcn 22692 cnmpt1plusg 22695 cnmpt2plusg 22696 tmdcn2 22697 submtmd 22712 tsmsadd 22755 mulrcn 22787 mhmhmeotmd 31170 xrge0pluscn 31183 |
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