Theorem nvclvec 24564

Description: A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
nvclvec	⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Proof of Theorem nvclvec

Step	Hyp	Ref	Expression
1		isnvc 24562	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simprbi 496	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2098  LVecclvec 20947  NrmModcnlm 24439  NrmVeccnvc 24440

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-nvc 24446

This theorem is referenced by:  nvctvc  24567  lssnvc  24569