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Theorem tgpcn 22622
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 22617 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
3 tgpcn.1 . . 3 𝐹 = (+𝑓𝐺)
42, 3tmdcn 22621 . 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 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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