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Theorem nvablo 22126
Description: The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvabl.1  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
nvablo  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )

Proof of Theorem nvablo
StepHypRef Expression
1 eqid 2442 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22125 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvabl.1 . . . 4  |-  G  =  ( +v `  U
)
43vafval 22113 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
54vcablo 22067 . 2  |-  ( ( 1st `  U )  e.  CVec OLD  ->  G  e. 
AbelOp )
62, 5syl 16 1  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   ` cfv 5483   1stc1st 6376   AbelOpcablo 21900   CVec
OLDcvc 22055   NrmCVeccnv 22094   +vcpv 22095
This theorem is referenced by:  nvgrp  22127  nvcom  22131  nvadd32  22134  nvadd4  22137  nvnnncan1  22160  nvaddsub  22171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-1st 6378  df-2nd 6379  df-vc 22056  df-nv 22102  df-va 22105  df-ba 22106  df-sm 22107  df-0v 22108  df-nmcv 22110
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