Theorem cphngp 25145

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

Step	Hyp	Ref	Expression
1		cphnlm 25144	. 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2		nlmngp 24638	. 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2098  NrmGrpcngp 24530  NrmModcnlm 24533  ℂPreHilccph 25138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fv 6557  df-ov 7422  df-nlm 24539  df-cph 25140

This theorem is referenced by:  cphnmf  25167  reipcl  25169  ipge0  25170  cphpyth  25188  cphipval2  25213  4cphipval2  25214  cphipval  25215  ipcn  25218  cnmpt1ip  25219  cnmpt2ip  25220  clsocv  25222  minveclem1  25396  minveclem2  25398  minveclem3b  25400  minveclem3  25401  minveclem4a  25402  minveclem4  25404  minveclem6  25406  minveclem7  25407  pjthlem1  25409  rrxngp  45811

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