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

Theorem tvclmod 23565
Description: A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tvclmod (𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Proof of Theorem tvclmod
StepHypRef Expression
1 tvctlm 23564 . 2 (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
2 tlmlmod 23556 . 2 (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  LModclmod 20336  TopModctlm 23525  TopVecctvc 23526
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
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-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-tlm 23529  df-tvc 23530
This theorem is referenced by:  tvclvec  23566
  Copyright terms: Public domain W3C validator