Theorem tgpcn 24070

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.)

Hypotheses

Ref	Expression
tgpcn.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgpcn.1	⊢ 𝐹 = (+𝑓‘𝐺)

Assertion

Ref	Expression
tgpcn	⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tgpcn

Step	Hyp	Ref	Expression
1		tgptmd 24065	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
3		tgpcn.1	. . 3 ⊢ 𝐹 = (+𝑓‘𝐺)
4	2, 3	tmdcn 24069	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
5	1, 4	syl 17	1 ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  ‘cfv 6488  (class class class)co 7359  TopOpenctopn 17379  +_𝑓cplusf 18600   Cn ccn 23210   ×_t ctx 23546  TopMndctmd 24056  TopGrpctgp 24057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5230

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-sbc 3725  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362  df-tmd 24058  df-tgp 24059

This theorem is referenced by:  pl1cn  34149