Theorem cphlvec 25142

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 25138	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		phllvec 21609	. 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2114  LVecclvec 21097  PreHilcphl 21604  ℂPreHilccph 25133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fv 6506  df-ov 7370  df-phl 21606  df-cph 25135

This theorem is referenced by:  cphnvc  25143  cphsubrg  25147  cphreccl  25148  cphqss  25155  hlprlem  25334  ishl2  25337