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Theorem cphngp 22781
 Description: A complex 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 22780 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 22291 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  NrmGrpcngp 22192  NrmModcnlm 22195  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-nlm 22201  df-cph 22776 This theorem is referenced by:  cphnmf  22803  reipcl  22805  ipge0  22806  cphipval2  22848  4cphipval2  22849  cphipval  22850  ipcn  22853  cnmpt1ip  22854  cnmpt2ip  22855  clsocv  22857  minveclem1  23003  minveclem2  23005  minveclem3b  23007  minveclem3  23008  minveclem4a  23009  minveclem4  23011  minveclem6  23013  minveclem7  23014  pjthlem1  23016  rrxngp  39178
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