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Theorem cphngp 25401
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 25400 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 24903 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 18 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wcel 2145  NrmGrpcngp 24803  NrmModcnlm 24806  ℂPreHilccph 25394
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fv 6541  df-ov 7416  df-nlm 24812  df-cph 25396
This theorem is used by:  cphnmf  25423  reipcl  25425  ipge0  25426  cphpyth  25444  cphipval2  25469  4cphipval2  25470  cphipval  25471  ipcn  25474  cnmpt1ip  25475  cnmpt2ip  25476  clsocv  25478  minveclem1  25652  minveclem2  25654  minveclem3b  25656  minveclem3  25657  minveclem4a  25658  minveclem4  25660  minveclem6  25662  minveclem7  25663  pjthlem1  25665  rrxngp  47113
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