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Theorem tmdmnd 18095
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 18094 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( + f `  G )  e.  ( ( (
TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp1bi 972 1  |-  ( G  e. TopMnd  ->  G  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5446  (class class class)co 6073   TopOpenctopn 13639   Mndcmnd 14674   + fcplusf 14677   TopSpctps 16951    Cn ccn 17278    tX ctx 17582  TopMndctmd 18090
This theorem is referenced by:  tmdmulg  18112  tmdgsum  18115  oppgtmd  18117  prdstmdd  18143  tsmsxp  18174  xrge0iifmhm  24315  esumcst  24445
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 2416  ax-nul 4330
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 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-tmd 18092
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