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Theorem cphngp 23782
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 23781 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 23287 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  NrmGrpcngp 23188  NrmModcnlm 23191  ℂPreHilccph 23775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fv 6336  df-ov 7142  df-nlm 23197  df-cph 23777
This theorem is referenced by:  cphnmf  23804  reipcl  23806  ipge0  23807  cphipval2  23849  4cphipval2  23850  cphipval  23851  ipcn  23854  cnmpt1ip  23855  cnmpt2ip  23856  clsocv  23858  minveclem1  24032  minveclem2  24034  minveclem3b  24036  minveclem3  24037  minveclem4a  24038  minveclem4  24040  minveclem6  24042  minveclem7  24043  pjthlem1  24045  rrxngp  42920
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