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Theorem tmdtps 23450
Description: A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdtps (๐บ โˆˆ TopMnd โ†’ ๐บ โˆˆ TopSp)

Proof of Theorem tmdtps
StepHypRef Expression
1 eqid 2733 . . 3 (+๐‘“โ€˜๐บ) = (+๐‘“โ€˜๐บ)
2 eqid 2733 . . 3 (TopOpenโ€˜๐บ) = (TopOpenโ€˜๐บ)
31, 2istmd 23448 . 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))))
43simp2bi 1147 1 (๐บ โˆˆ TopMnd โ†’ ๐บ โˆˆ TopSp)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2107  โ€˜cfv 6500  (class class class)co 7361  TopOpenctopn 17311  +๐‘“cplusf 18502  Mndcmnd 18564  TopSpctps 22304   Cn ccn 22598   ร—t ctx 22934  TopMndctmd 23444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-tmd 23446
This theorem is referenced by:  tgptps  23454  tmdtopon  23455  submtmd  23478  prdstmdd  23498  tsmsadd  23521  tsmssplit  23526  tlmtps  23562
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