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Theorem tgpcn 18035
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  e.  TopGrp  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )

Proof of Theorem tgpcn
StepHypRef Expression
1 tgptmd 18030 . 2  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
2 tgpcn.j . . 3  |-  J  =  ( TopOpen `  G )
3 tgpcn.1 . . 3  |-  F  =  ( + f `  G )
42, 3tmdcn 18034 . 2  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
51, 4syl 16 1  |-  ( G  e.  TopGrp  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   TopOpenctopn 13576   + fcplusf 14614    Cn ccn 17210    tX ctx 17513  TopMndctmd 18021   TopGrpctgp 18022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-tmd 18023  df-tgp 18024
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