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Theorem tmdcn 20312
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 20303 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J
) ) )
43simp3bi 1008 1  |-  ( G  e. TopMnd  ->  F  e.  ( ( J  tX  J
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   TopOpenctopn 14668   Mndcmnd 15717   +fcplusf 15720   TopSpctps 19159    Cn ccn 19486    tX ctx 19791  TopMndctmd 20299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-tmd 20301
This theorem is referenced by:  tgpcn  20313  cnmpt1plusg  20316  cnmpt2plusg  20317  tmdcn2  20318  submtmd  20333  tsmsadd  20379  mulrcn  20411  mhmhmeotmd  27533  xrge0pluscn  27546
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