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

Theorem tlmtps 24105
Description: A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tlmtps (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Proof of Theorem tlmtps
StepHypRef Expression
1 tlmtmd 24104 . 2 (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
2 tmdtps 23993 . 2 (𝑊 ∈ TopMnd → 𝑊 ∈ TopSp)
31, 2syl 17 1 (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  TopSpctps 22847  TopMndctmd 23987  TopModctlm 24075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tmd 23989  df-tlm 24079
This theorem is referenced by:  cnmpt1vsca  24111  cnmpt2vsca  24112  tlmtgp  24113
  Copyright terms: Public domain W3C validator