Theorem tvclmod 24120

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 24119	. 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
2		tlmlmod 24111	. 2 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Syntax hints:   → wi 4   ∈ wcel 2098  LModclmod 20748  TopModctlm 24080  TopVecctvc 24081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427  df-tlm 24084  df-tvc 24085

This theorem is referenced by:  tvclvec  24121