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Theorem tmdcn 20455
Description: In a topological monoid, the operation  F representing the functionalization of the operator slot  +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
tgpcn.j  |-  J  =  ( TopOpen `  G )
tgpcn.1  |-  F  =  ( +f `  G )
Assertion
Ref Expression
tmdcn  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )

Proof of Theorem tmdcn
StepHypRef Expression
1 tgpcn.1 . . 3  |-  F  =  ( +f `  G )
2 tgpcn.j . . 3  |-  J  =  ( TopOpen `  G )
31, 2istmd 20446 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
43simp3bi 1014 1  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   TopOpenctopn 14696   +fcplusf 15743   Mndcmnd 15793   TopSpctps 19270    Cn ccn 19598    tX ctx 19934  TopMndctmd 20442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-tmd 20444
This theorem is referenced by:  tgpcn  20456  cnmpt1plusg  20459  cnmpt2plusg  20460  tmdcn2  20461  submtmd  20476  tsmsadd  20522  mulrcn  20554  mhmhmeotmd  27782  xrge0pluscn  27795
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