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Theorem nvgrp 22088
Description: The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvabl.1  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
nvgrp  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )

Proof of Theorem nvgrp
StepHypRef Expression
1 nvabl.1 . . 3  |-  G  =  ( +v `  U
)
21nvablo 22087 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
3 ablogrpo 21864 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 16 1  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446   GrpOpcgr 21766   AbelOpcablo 21861   NrmCVeccnv 22055   +vcpv 22056
This theorem is referenced by:  nvgf  22089  nvgcl  22091  nvass  22093  nvrcan  22096  nvlcan  22097  nvzcl  22107  nv0rid  22108  nv0lid  22109  nvinvfval  22113  nvmval  22115  nvmfval  22117  nvnnncan2  22122  nvnegneg  22124  nvrinv  22126  nvlinv  22127  nvaddsubass  22131  nvmtri2  22153  hhshsslem1  22759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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