MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tmdtps Unicode version

Theorem tmdtps 17759
Description: A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tmdtps  |-  ( G  e. TopMnd  ->  G  e.  TopSp )

Proof of Theorem tmdtps
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( + f `  G )  =  ( + f `  G )
2 eqid 2283 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 17757 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( + f `  G )  e.  ( ( (
TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp2bi 971 1  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   ` cfv 5255  (class class class)co 5858   TopOpenctopn 13326   Mndcmnd 14361   + fcplusf 14364   TopSpctps 16634    Cn ccn 16954    tX ctx 17255  TopMndctmd 17753
This theorem is referenced by:  tgptps  17763  tmdtopon  17764  submtmd  17787  prdstmdd  17806  tsmsadd  17829  tsmssplit  17834  tlmtps  17870  esumf1o  23429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-tmd 17755
  Copyright terms: Public domain W3C validator