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Theorem vc2 25649
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 25645 . . 3  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
54, 4oveq12d 6288 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 1 S A ) )  =  ( A G A ) )
6 df-2 10590 . . . 4  |-  2  =  ( 1  +  1 )
76oveq1i 6280 . . 3  |-  ( 2 S A )  =  ( ( 1  +  1 ) S A )
8 ax-1cn 9539 . . . 4  |-  1  e.  CC
91, 2, 3vcdir 25647 . . . . 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 1319 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( 1  e.  CC  /\  A  e.  X ) )  ->  ( (
1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
118, 10mpanr1 681 . . 3  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
127, 11syl5req 2508 . 2  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( ( 1 S A ) G ( 1 S A ) )  =  ( 2 S A ) )
135, 12eqtr3d 2497 1  |-  ( ( W  e.  CVecOLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   CCcc 9479   1c1 9482    + caddc 9484   2c2 10581   CVecOLDcvc 25639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-1cn 9539
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774  df-2 10590  df-vc 25640
This theorem is referenced by:  nv2  25728  ipdirilem  25945
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