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Theorem cphngp 23704
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 23703 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 23213 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  NrmGrpcngp 23114  NrmModcnlm 23117  ℂPreHilccph 23697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7148  df-nlm 23123  df-cph 23699
This theorem is referenced by:  cphnmf  23726  reipcl  23728  ipge0  23729  cphipval2  23771  4cphipval2  23772  cphipval  23773  ipcn  23776  cnmpt1ip  23777  cnmpt2ip  23778  clsocv  23780  minveclem1  23954  minveclem2  23956  minveclem3b  23958  minveclem3  23959  minveclem4a  23960  minveclem4  23962  minveclem6  23964  minveclem7  23965  pjthlem1  23967  rrxngp  42447
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