Theorem nvclvec 24718

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 24716	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simprbi 496	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2108  LVecclvec 21101  NrmModcnlm 24593  NrmVeccnvc 24594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-nvc 24600

This theorem is referenced by:  nvctvc  24721  lssnvc  24723