Theorem cphlvec 25217

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 25213	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2		phllvec 21661	. 2 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Syntax hints:   → wi 4   ∈ wcel 2141  LVecclvec 21149  PreHilcphl 21656  ℂPreHilccph 25208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fv 6525  df-ov 7395  df-phl 21658  df-cph 25210

This theorem is referenced by:  cphnvc  25218  cphsubrg  25222  cphreccl  25223  cphqss  25230  hlprlem  25409  ishl2  25412