Theorem cphnvc 22976

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 22972	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2		cphlvec 22975	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3		isnvc 22499	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
4	1, 2, 3	sylanbrc 698	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Syntax hints:   → wi 4   ∈ wcel 1990  LVecclvec 19102  NrmModcnlm 22385  NrmVeccnvc 22386  ℂPreHilccph 22966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-phl 19971  df-nvc 22392  df-cph 22968

This theorem is referenced by:  ishl2  23166