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Theorem vc2 24065
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
vci.1  |-  G  =  ( 1st `  W
)
vci.2  |-  S  =  ( 2nd `  W
)
vci.3  |-  X  =  ran  G
Assertion
Ref Expression
vc2  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )

Proof of Theorem vc2
StepHypRef Expression
1 vci.1 . . . 4  |-  G  =  ( 1st `  W
)
2 vci.2 . . . 4  |-  S  =  ( 2nd `  W
)
3 vci.3 . . . 4  |-  X  =  ran  G
41, 2, 3vcid 24061 . . 3  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
54, 4oveq12d 6205 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 1 S A ) )  =  ( A G A ) )
6 df-2 10478 . . . 4  |-  2  =  ( 1  +  1 )
76oveq1i 6197 . . 3  |-  ( 2 S A )  =  ( ( 1  +  1 ) S A )
8 ax-1cn 9438 . . . 4  |-  1  e.  CC
91, 2, 3vcdir 24063 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  X )
)  ->  ( (
1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
108, 9mp3anr1 1312 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( 1  e.  CC  /\  A  e.  X ) )  ->  ( (
1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
118, 10mpanr1 683 . . 3  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
127, 11syl5req 2504 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 1 S A ) )  =  ( 2 S A ) )
135, 12eqtr3d 2493 1  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ran crn 4936   ` cfv 5513  (class class class)co 6187   1stc1st 6672   2ndc2nd 6673   CCcc 9378   1c1 9381    + caddc 9383   2c2 10469   CVecOLDcvc 24055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-1cn 9438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-ov 6190  df-1st 6674  df-2nd 6675  df-2 10478  df-vc 24056
This theorem is referenced by:  nv2  24144  ipdirilem  24361
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