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Theorem tgpcn 23588
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 = (TopOpenβ€˜πΊ)
tgpcn.1 𝐹 = (+π‘“β€˜πΊ)
Assertion
Ref Expression
tgpcn (𝐺 ∈ TopGrp β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))

Proof of Theorem tgpcn
StepHypRef Expression
1 tgptmd 23583 . 2 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
2 tgpcn.j . . 3 𝐽 = (TopOpenβ€˜πΊ)
3 tgpcn.1 . . 3 𝐹 = (+π‘“β€˜πΊ)
42, 3tmdcn 23587 . 2 (𝐺 ∈ TopMnd β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
51, 4syl 17 1 (𝐺 ∈ TopGrp β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  TopOpenctopn 17367  +𝑓cplusf 18558   Cn ccn 22728   Γ—t ctx 23064  TopMndctmd 23574  TopGrpctgp 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-tmd 23576  df-tgp 23577
This theorem is referenced by:  pl1cn  32935
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