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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 23574 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
2 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | tgpcn.1 | . . 3 ⊢ 𝐹 = (+𝑓‘𝐺) | |
4 | 2, 3 | tmdcn 23578 | . 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 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 TopOpenctopn 17363 +𝑓cplusf 18554 Cn ccn 22719 ×t ctx 23055 TopMndctmd 23565 TopGrpctgp 23566 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 df-tmd 23567 df-tgp 23568 |
This theorem is referenced by: pl1cn 32923 |
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