Theorem cphnvc 23463

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 23459	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2		cphlvec 23462	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3		isnvc 22987	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
4	1, 2, 3	sylanbrc 583	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 2081  LVecclvec 19564  NrmModcnlm 22873  NrmVeccnvc 22874  ℂPreHilccph 23453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fv 6233  df-ov 7019  df-phl 20452  df-nvc 22880  df-cph 23455

This theorem is referenced by:  ishl2  23656  csschl  23662