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Theorem cphngp 23019
Description: A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cphngp (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)

Proof of Theorem cphngp
StepHypRef Expression
1 cphnlm 23018 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 22528 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  NrmGrpcngp 22429  NrmModcnlm 22432  ℂPreHilccph 23012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-ov 6693  df-nlm 22438  df-cph 23014
This theorem is referenced by:  cphnmf  23041  reipcl  23043  ipge0  23044  cphipval2  23086  4cphipval2  23087  cphipval  23088  ipcn  23091  cnmpt1ip  23092  cnmpt2ip  23093  clsocv  23095  minveclem1  23241  minveclem2  23243  minveclem3b  23245  minveclem3  23246  minveclem4a  23247  minveclem4  23249  minveclem6  23251  minveclem7  23252  pjthlem1  23254  rrxngp  40820
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