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Mirrors > Home > MPE Home > Th. List > tgpcn | Structured version Visualization version GIF version |
Description: In a topological group, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpcn.1 | ⊢ 𝐹 = (+𝑓‘𝐺) |
Ref | Expression |
---|---|
tgpcn | ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptmd 22617 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | tgpcn.1 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | tmdcn 22621 | . 2 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 TopOpenctopn 16685 +𝑓cplusf 17839 Cn ccn 21762 ×t ctx 22098 TopMndctmd 22608 TopGrpctgp 22609 |
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 2793 ax-nul 5202 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7148 df-tmd 22610 df-tgp 22611 |
This theorem is referenced by: pl1cn 31098 |
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