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Theorem tmdcn 18113
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 18104 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
43simp3bi 974 1  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   TopOpenctopn 13649   Mndcmnd 14684   + fcplusf 14687   TopSpctps 16961    Cn ccn 17288    tX ctx 17592  TopMndctmd 18100
This theorem is referenced by:  tgpcn  18114  cnmpt1plusg  18117  cnmpt2plusg  18118  tmdcn2  18119  submtmd  18134  tsmsadd  18176  mulrcn  18208  mhmhmeotmd  24313  xrge0pluscn  24326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-tmd 18102
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