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

Theorem tmdtps 18028
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 2388 . . 3  |-  ( + f `  G )  =  ( + f `  G )
2 eqid 2388 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 18026 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( + f `  G )  e.  ( ( (
TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp2bi 973 1  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   ` cfv 5395  (class class class)co 6021   TopOpenctopn 13577   Mndcmnd 14612   + fcplusf 14615   TopSpctps 16885    Cn ccn 17211    tX ctx 17514  TopMndctmd 18022
This theorem is referenced by:  tgptps  18032  tmdtopon  18033  submtmd  18056  prdstmdd  18075  tsmsadd  18098  tsmssplit  18103  tlmtps  18139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-tmd 18024
  Copyright terms: Public domain W3C validator