Theorem tgpcn 21888

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 = (+𝑓‘𝐺)

Assertion

Ref	Expression
tgpcn	⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Proof of Theorem tgpcn

Step	Hyp	Ref	Expression
1		tgptmd 21883	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
3		tgpcn.1	. . 3 ⊢ 𝐹 = (+𝑓‘𝐺)
4	2, 3	tmdcn 21887	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
5	1, 4	syl 17	1 ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ‘cfv 5888  (class class class)co 6650  TopOpenctopn 16082  +_𝑓cplusf 17239   Cn ccn 21028   ×_t ctx 21363  TopMndctmd 21874  TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  pl1cn  30001