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

Theorem tmdtps 20701
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 2457 . . 3  |-  ( +f `  G )  =  ( +f `  G )
2 eqid 2457 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
31, 2istmd 20699 . 2  |-  ( G  e. TopMnd 
<->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  ( +f `  G
)  e.  ( ( ( TopOpen `  G )  tX  ( TopOpen `  G )
)  Cn  ( TopOpen `  G ) ) ) )
43simp2bi 1012 1  |-  ( G  e. TopMnd  ->  G  e.  TopSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819   ` cfv 5594  (class class class)co 6296   TopOpenctopn 14839   +fcplusf 15996   Mndcmnd 16046   TopSpctps 19524    Cn ccn 19852    tX ctx 20187  TopMndctmd 20695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-tmd 20697
This theorem is referenced by:  tgptps  20705  tmdtopon  20706  submtmd  20729  prdstmdd  20748  tsmsadd  20775  tsmssplit  20780  tlmtps  20816
  Copyright terms: Public domain W3C validator