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Theorem tmdmnd 18058
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 2404 . . 3  |-  ( + f `  G )  =  ( + f `  G )
2 eqid 2404 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 18057 . 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 1721   ` cfv 5413  (class class class)co 6040   TopOpenctopn 13604   Mndcmnd 14639   + fcplusf 14642   TopSpctps 16916    Cn ccn 17242    tX ctx 17545  TopMndctmd 18053
This theorem is referenced by:  tmdmulg  18075  tmdgsum  18078  oppgtmd  18080  prdstmdd  18106  tsmsxp  18137  xrge0iifmhm  24278  esumcst  24408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-tmd 18055
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