Theorem cphnvc 24693

Description: A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)

Assertion

Ref	Expression
cphnvc	⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Proof of Theorem cphnvc

Step	Hyp	Ref	Expression
1		cphnlm 24689	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2		cphlvec 24692	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3		isnvc 24212	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
4	1, 2, 3	sylanbrc 584	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2107  LVecclvec 20713  NrmModcnlm 24089  NrmVeccnvc 24090  ℂPreHilccph 24683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-phl 21179  df-nvc 24096  df-cph 24685

This theorem is referenced by:  ishl2  24887  csschl  24893