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Theorem tmdtps 23579
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 2732 . . 3 (+๐‘“โ€˜๐บ) = (+๐‘“โ€˜๐บ)
2 eqid 2732 . . 3 (TopOpenโ€˜๐บ) = (TopOpenโ€˜๐บ)
31, 2istmd 23577 . 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 1146 1 (๐บ โˆˆ TopMnd โ†’ ๐บ โˆˆ TopSp)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2106  โ€˜cfv 6543  (class class class)co 7408  TopOpenctopn 17366  +๐‘“cplusf 18557  Mndcmnd 18624  TopSpctps 22433   Cn ccn 22727   ร—t ctx 23063  TopMndctmd 23573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-tmd 23575
This theorem is referenced by:  tgptps  23583  tmdtopon  23584  submtmd  23607  prdstmdd  23627  tsmsadd  23650  tsmssplit  23655  tlmtps  23691
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