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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 24082 | . 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))) |
| 4 | 3 | simp3bi 1148 | 1 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 TopOpenctopn 17466 +𝑓cplusf 18650 Mndcmnd 18747 TopSpctps 22938 Cn ccn 23232 ×t ctx 23568 TopMndctmd 24078 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tmd 24080 |
| This theorem is referenced by: tgpcn 24092 cnmpt1plusg 24095 cnmpt2plusg 24096 tmdcn2 24097 submtmd 24112 tsmsadd 24155 mulrcn 24187 mhmhmeotmd 33926 xrge0pluscn 33939 |
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