Theorem nvclvec 22501

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

Assertion

Ref	Expression
nvclvec	|- ( W e. NrmVec -> W e. LVec )

Proof of Theorem nvclvec

Step	Hyp	Ref	Expression
1		isnvc 22499	. 2 |- ( W e. NrmVec <-> ( W e. NrmMod /\ W e. LVec ) )
2	1	simprbi 480	1 |- ( W e. NrmVec -> W e. LVec )

Syntax hints:   -> wi 4   e. wcel 1990   LVecclvec 19102  NrmModcnlm 22385  NrmVeccnvc 22386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-nvc 22392

This theorem is referenced by:  nvctvc  22504  lssnvc  22506