Theorem nvclvec 23300

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 23298	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simprbi 499	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2110  LVecclvec 19868  NrmModcnlm 23184  NrmVeccnvc 23185

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-in 3943  df-nvc 23191

This theorem is referenced by:  nvctvc  23303  lssnvc  23305