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Theorem tmdmnd 19490
Description: A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdmnd  |-  ( G  e. TopMnd  ->  G  e.  Mnd )

Proof of Theorem tmdmnd
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( +f `  G )  =  ( +f `  G )
2 eqid 2435 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 19489 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( +f `  G
)  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp1bi 998 1  |-  ( G  e. TopMnd  ->  G  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1757   ` cfv 5408  (class class class)co 6082   TopOpenctopn 14345   Mndcmnd 15394   +fcplusf 15397   TopSpctps 18345    Cn ccn 18672    tX ctx 18977  TopMndctmd 19485
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 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-nul 4411
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2966  df-sbc 3178  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-br 4283  df-iota 5371  df-fv 5416  df-ov 6085  df-tmd 19487
This theorem is referenced by:  tmdmulg  19507  tmdgsum  19510  oppgtmd  19512  prdstmdd  19538  tsmsxp  19573  xrge0iifmhm  26225  esumcst  26370
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