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Theorem nvcom 22092
Description: The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvgcl.1  |-  X  =  ( BaseSet `  U )
nvgcl.2  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
nvcom  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )

Proof of Theorem nvcom
StepHypRef Expression
1 nvgcl.2 . . 3  |-  G  =  ( +v `  U
)
21nvablo 22087 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
3 nvgcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 22075 . . 3  |-  X  =  ran  G
54ablocom 21865 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
62, 5syl3an1 1217 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   AbelOpcablo 21861   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057
This theorem is referenced by:  nvadd12  22094  nvmval2  22116  nvpncan  22130  nvsubsub23  22135  nvdif  22146  nvpi  22147  nvabs  22154  dipcj  22205  hlcom  22394
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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