MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgptps Structured version   Visualization version   GIF version

Theorem tgptps 23454
Description: A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
tgptps (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Proof of Theorem tgptps
StepHypRef Expression
1 tgptmd 23453 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tmdtps 23450 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  TopSpctps 22304  TopMndctmd 23444  TopGrpctgp 23445
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  df-tgp 23447
This theorem is referenced by:  tgptopon  23456  istgp2  23465  tsmsinv  23522  tsmssub  23523  tgptsmscls  23524  tgptsmscld  23525  tsmsxplem1  23527  tsmsxp  23529  trgtps  23544  nrgtrg  24077
  Copyright terms: Public domain W3C validator