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Theorem tgpcn 23587
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 23582 . 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 23586 . 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 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7408  TopOpenctopn 17366  +𝑓cplusf 18557   Cn ccn 22727   Γ—t ctx 23063  TopMndctmd 23573  TopGrpctgp 23574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-tmd 23575  df-tgp 23576
This theorem is referenced by:  pl1cn  32930
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