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Theorem tmdmnd 20442
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 2467 . . 3  |-  ( +f `  G )  =  ( +f `  G )
2 eqid 2467 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 20441 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( +f `  G
)  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp1bi 1011 1  |-  ( G  e. TopMnd  ->  G  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5594  (class class class)co 6295   TopOpenctopn 14694   +fcplusf 15743   Mndcmnd 15793   TopSpctps 19266    Cn ccn 19593    tX ctx 19929  TopMndctmd 20437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-tmd 20439
This theorem is referenced by:  tmdmulg  20459  tmdgsum  20462  oppgtmd  20464  prdstmdd  20490  tsmsxp  20525  xrge0iifmhm  27746  esumcst  27896
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