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Theorem tmdmnd 17774
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 2296 . . 3  |-  ( + f `  G )  =  ( + f `  G )
2 eqid 2296 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 17773 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( + f `  G )  e.  ( ( (
TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp1bi 970 1  |-  ( G  e. TopMnd  ->  G  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   ` cfv 5271  (class class class)co 5874   TopOpenctopn 13342   Mndcmnd 14377   + fcplusf 14380   TopSpctps 16650    Cn ccn 16970    tX ctx 17271  TopMndctmd 17769
This theorem is referenced by:  tmdmulg  17791  tmdgsum  17794  oppgtmd  17796  prdstmdd  17822  tsmsxp  17853  xrge0iifmhm  23336  esumcst  23451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-tmd 17771
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