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Theorem tgpcn 21698
 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 21693 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
3 tgpcn.1 . . 3 𝐹 = (+𝑓𝐺)
42, 3tmdcn 21697 . 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 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  TopOpenctopn 15905  +𝑓cplusf 17062   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684  TopGrpctgp 21685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tmd 21686  df-tgp 21687 This theorem is referenced by:  pl1cn  29329
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