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Theorem nvgrp 28016
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 = ( +𝑣𝑈)
Assertion
Ref Expression
nvgrp (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)

Proof of Theorem nvgrp
StepHypRef Expression
1 nvabl.1 . . 3 𝐺 = ( +𝑣𝑈)
21nvablo 28015 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
3 ablogrpo 27946 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  cfv 6123  GrpOpcgr 27888  AbelOpcablo 27943  NrmCVeccnv 27983   +𝑣 cpv 27984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-1st 7428  df-2nd 7429  df-ablo 27944  df-vc 27958  df-nv 27991  df-va 27994  df-ba 27995  df-sm 27996  df-0v 27997  df-nmcv 27999
This theorem is referenced by:  nvgf  28017  nvgcl  28019  nvass  28021  nvrcan  28023  nvzcl  28033  nv0rid  28034  nv0lid  28035  nvinvfval  28039  nvmval  28041  nvmfval  28043  nvnegneg  28048  nvrinv  28050  nvlinv  28051  hhshsslem1  28668
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