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Theorem tgpcn 21935
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 = (+𝑓𝐺)
Assertion
Ref Expression
tgpcn (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tgpcn
StepHypRef Expression
1 tgptmd 21930 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
3 tgpcn.1 . . 3 𝐹 = (+𝑓𝐺)
42, 3tmdcn 21934 . 2 (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
51, 4syl 17 1 (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  TopOpenctopn 16129  +𝑓cplusf 17286   Cn ccn 21076   ×t ctx 21411  TopMndctmd 21921  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-tmd 21923  df-tgp 21924
This theorem is referenced by:  pl1cn  30129
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