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Theorem cphnvc 22784
 Description: A complex 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 22780 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 cphlvec 22783 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
3 isnvc 22309 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
41, 2, 3sylanbrc 695 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  LVecclvec 18923  NrmModcnlm 22195  NrmVeccnvc 22196  ℂPreHilccph 22774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-phl 19790  df-nvc 22202  df-cph 22776 This theorem is referenced by:  ishl2  22974
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