Theorem nvclvec 24865

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 24863	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simprbi 502	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  LVecclvec 21234  NrmModcnlm 24748  NrmVeccnvc 24749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-nvc 24755

This theorem is used by:  nvctvc  24868  lssnvc  24870