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Theorem tvclmod 22512
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 22511 . 2 (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
2 tlmlmod 22503 . 2 (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2050  LModclmod 19359  TopModctlm 22472  TopVecctvc 22473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-iota 6154  df-fv 6198  df-ov 6981  df-tlm 22476  df-tvc 22477
This theorem is referenced by:  tvclvec  22513
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