Theorem tvclmod 24222

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

Step	Hyp	Ref	Expression
1		tvctlm 24221	. 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
2		tlmlmod 24213	. 2 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2106  LModclmod 20875  TopModctlm 24182  TopVecctvc 24183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tlm 24186  df-tvc 24187

This theorem is referenced by:  tvclvec  24223