Theorem tgpcn 24209

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 24204	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
3		tgpcn.1	. . 3 ⊢ 𝐹 = (+𝑓‘𝐺)
4	2, 3	tmdcn 24208	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
5	1, 4	syl 18	1 ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  TopOpenctopn 17473  +_𝑓cplusf 18694   Cn ccn 23349   ×_t ctx 23685  TopMndctmd 24195  TopGrpctgp 24196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-tmd 24197  df-tgp 24198

This theorem is referenced by:  pl1cn  34289