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Theorem cphnvc 23870
 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
StepHypRef Expression
1 cphnlm 23866 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 cphlvec 23869 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3 isnvc 23390 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
41, 2, 3sylanbrc 587 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  LVecclvec 19935  NrmModcnlm 23275  NrmVeccnvc 23276  ℂPreHilccph 23860 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fv 6344  df-ov 7154  df-phl 20384  df-nvc 23282  df-cph 23862 This theorem is referenced by:  ishl2  24063  csschl  24069
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