MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvablo Unicode version

Theorem nvablo 21286
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 2358 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21285 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvabl.1 . . . 4  |-  G  =  ( +v `  U
)
43vafval 21273 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
54vcablo 21227 . 2  |-  ( ( 1st `  U )  e.  CVec OLD  ->  G  e. 
AbelOp )
62, 5syl 15 1  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   ` cfv 5337   1stc1st 6207   AbelOpcablo 21060   CVec
OLDcvc 21215   NrmCVeccnv 21254   +vcpv 21255
This theorem is referenced by:  nvgrp  21287  nvcom  21291  nvadd32  21294  nvadd4  21297  nvnnncan1  21320  nvaddsub  21331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-1st 6209  df-2nd 6210  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-nmcv 21270
  Copyright terms: Public domain W3C validator