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Theorem tmdmnd 19788
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 2454 . . 3  |-  ( +f `  G )  =  ( +f `  G )
2 eqid 2454 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 19787 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( +f `  G
)  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp1bi 1003 1  |-  ( G  e. TopMnd  ->  G  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   ` cfv 5529  (class class class)co 6203   TopOpenctopn 14483   Mndcmnd 15532   +fcplusf 15535   TopSpctps 18643    Cn ccn 18970    tX ctx 19275  TopMndctmd 19783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-tmd 19785
This theorem is referenced by:  tmdmulg  19805  tmdgsum  19808  oppgtmd  19810  prdstmdd  19836  tsmsxp  19871  xrge0iifmhm  26537  esumcst  26682
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