Theorem cphlvec 25082

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 25078	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		phllvec 21545	. 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2109  LVecclvec 21016  PreHilcphl 21540  ℂPreHilccph 25073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fv 6522  df-ov 7393  df-phl 21542  df-cph 25075

This theorem is referenced by:  cphnvc  25083  cphsubrg  25087  cphreccl  25088  cphqss  25095  hlprlem  25274  ishl2  25277