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Theorem cphngp 25433
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 25432 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmngp 24935 . 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 24835  NrmModcnlm 24838  ℂPreHilccph 25426
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 6491  df-fv 6543  df-ov 7419  df-nlm 24844  df-cph 25428
This theorem is used by:  cphnmf  25455  reipcl  25457  ipge0  25458  cphpyth  25476  cphipval2  25501  4cphipval2  25502  cphipval  25503  ipcn  25506  cnmpt1ip  25507  cnmpt2ip  25508  clsocv  25510  minveclem1  25684  minveclem2  25686  minveclem3b  25688  minveclem3  25689  minveclem4a  25690  minveclem4  25692  minveclem6  25694  minveclem7  25695  pjthlem1  25697  rrxngp  47178
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