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Theorem vcsub4 23952
Description: Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
vcni.1  |-  G  =  ( 1st `  W
)
vcni.2  |-  S  =  ( 2nd `  W
)
vcni.3  |-  X  =  ran  G
Assertion
Ref Expression
vcsub4  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( A G B ) G (
-u 1 S ( C G D ) ) )  =  ( ( A G (
-u 1 S C ) ) G ( B G ( -u
1 S D ) ) ) )

Proof of Theorem vcsub4
StepHypRef Expression
1 neg1cn 10423 . . . . 5  |-  -u 1  e.  CC
2 vcni.1 . . . . . 6  |-  G  =  ( 1st `  W
)
3 vcni.2 . . . . . 6  |-  S  =  ( 2nd `  W
)
4 vcni.3 . . . . . 6  |-  X  =  ran  G
52, 3, 4vcdi 23928 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  ( -u 1  e.  CC  /\  C  e.  X  /\  D  e.  X ) )  -> 
( -u 1 S ( C G D ) )  =  ( (
-u 1 S C ) G ( -u
1 S D ) ) )
61, 5mp3anr1 1311 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( -u 1 S ( C G D ) )  =  ( ( -u
1 S C ) G ( -u 1 S D ) ) )
763adant2 1007 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( -u 1 S ( C G D ) )  =  ( (
-u 1 S C ) G ( -u
1 S D ) ) )
87oveq2d 6105 . 2  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( A G B ) G (
-u 1 S ( C G D ) ) )  =  ( ( A G B ) G ( (
-u 1 S C ) G ( -u
1 S D ) ) ) )
92, 3, 4vccl 23926 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X
)
101, 9mp3an2 1302 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X
)
112, 3, 4vccl 23926 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  -u 1  e.  CC  /\  D  e.  X )  ->  ( -u 1 S D )  e.  X
)
121, 11mp3an2 1302 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  D  e.  X )  ->  ( -u 1 S D )  e.  X
)
1310, 12anim12dan 833 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( -u 1 S C )  e.  X  /\  ( -u 1 S D )  e.  X ) )
14133adant2 1007 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( -u 1 S C )  e.  X  /\  ( -u 1 S D )  e.  X
) )
152, 4vca4 23939 . . 3  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( ( -u 1 S C )  e.  X  /\  ( -u 1 S D )  e.  X ) )  ->  ( ( A G B ) G ( ( -u 1 S C ) G (
-u 1 S D ) ) )  =  ( ( A G ( -u 1 S C ) ) G ( B G (
-u 1 S D ) ) ) )
1614, 15syld3an3 1263 . 2  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( A G B ) G ( ( -u 1 S C ) G (
-u 1 S D ) ) )  =  ( ( A G ( -u 1 S C ) ) G ( B G (
-u 1 S D ) ) ) )
178, 16eqtrd 2473 1  |-  ( ( W  e.  CVecOLD  /\  ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( ( A G B ) G (
-u 1 S ( C G D ) ) )  =  ( ( A G (
-u 1 S C ) ) G ( B G ( -u
1 S D ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4839   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574   CCcc 9278   1c1 9281   -ucneg 9594   CVecOLDcvc 23921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-ltxr 9421  df-sub 9595  df-neg 9596  df-grpo 23676  df-ablo 23767  df-vc 23922
This theorem is referenced by: (None)
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