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Theorem tmdmnd 18026
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 2387 . . 3  |-  ( + f `  G )  =  ( + f `  G )
2 eqid 2387 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 18025 . 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 1717   ` cfv 5394  (class class class)co 6020   TopOpenctopn 13576   Mndcmnd 14611   + fcplusf 14614   TopSpctps 16884    Cn ccn 17210    tX ctx 17513  TopMndctmd 18021
This theorem is referenced by:  tmdmulg  18043  tmdgsum  18046  oppgtmd  18048  prdstmdd  18074  tsmsxp  18105  xrge0iifmhm  24129  esumcst  24251
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279
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 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-tmd 18023
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