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Theorem tmdmnd 23992
Description: A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdmnd (๐บ โˆˆ TopMnd โ†’ ๐บ โˆˆ Mnd)

Proof of Theorem tmdmnd
StepHypRef Expression
1 eqid 2728 . . 3 (+๐‘“โ€˜๐บ) = (+๐‘“โ€˜๐บ)
2 eqid 2728 . . 3 (TopOpenโ€˜๐บ) = (TopOpenโ€˜๐บ)
31, 2istmd 23991 . 2 (๐บ โˆˆ TopMnd โ†” (๐บ โˆˆ Mnd โˆง ๐บ โˆˆ TopSp โˆง (+๐‘“โ€˜๐บ) โˆˆ (((TopOpenโ€˜๐บ) ร—t (TopOpenโ€˜๐บ)) Cn (TopOpenโ€˜๐บ))))
43simp1bi 1143 1 (๐บ โˆˆ TopMnd โ†’ ๐บ โˆˆ Mnd)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2099  โ€˜cfv 6548  (class class class)co 7420  TopOpenctopn 17403  +๐‘“cplusf 18597  Mndcmnd 18694  TopSpctps 22847   Cn ccn 23141   ร—t ctx 23477  TopMndctmd 23987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tmd 23989
This theorem is referenced by:  tmdmulg  24009  tmdgsum  24012  oppgtmd  24014  prdstmdd  24041  tsmsxp  24072  xrge0iifmhm  33540  esumcst  33682
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