Theorem cphnvc 25231

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 25227	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2		cphlvec 25230	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3		isnvc 24739	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
4	1, 2, 3	sylanbrc 582	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2108  LVecclvec 21126  NrmModcnlm 24616  NrmVeccnvc 24617  ℂPreHilccph 25221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fv 6583  df-ov 7453  df-phl 21669  df-nvc 24623  df-cph 25223

This theorem is referenced by:  ishl2  25425  csschl  25431