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Theorem tmdcn 19653
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 19644 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
43simp3bi 1005 1  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090   TopOpenctopn 14359   Mndcmnd 15408   +fcplusf 15411   TopSpctps 18500    Cn ccn 18827    tX ctx 19132  TopMndctmd 19640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4420
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093  df-tmd 19642
This theorem is referenced by:  tgpcn  19654  cnmpt1plusg  19657  cnmpt2plusg  19658  tmdcn2  19659  submtmd  19674  tsmsadd  19720  mulrcn  19752  mhmhmeotmd  26356  xrge0pluscn  26369
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