Theorem cphlvec 25147

Description: A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Assertion

Ref	Expression
cphlvec	⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Proof of Theorem cphlvec

Step	Hyp	Ref	Expression
1		cphphl 25143	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		phllvec 21578	. 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2098  LVecclvec 20999  PreHilcphl 21573  ℂPreHilccph 25138

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 2696  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fv 6557  df-ov 7422  df-phl 21575  df-cph 25140

This theorem is referenced by:  cphnvc  25148  cphsubrg  25152  cphreccl  25153  cphqss  25160  hlprlem  25339  ishl2  25342