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Theorem vcsub4 26040
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 10713 . . . . 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 26016 . . . . 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 1357 . . . 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 1024 . . 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 6321 . 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 26014 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X
)
101, 9mp3an2 1348 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  C  e.  X )  ->  ( -u 1 S C )  e.  X
)
112, 3, 4vccl 26014 . . . . . 6  |-  ( ( W  e.  CVecOLD  /\  -u 1  e.  CC  /\  D  e.  X )  ->  ( -u 1 S D )  e.  X
)
121, 11mp3an2 1348 . . . . 5  |-  ( ( W  e.  CVecOLD  /\  D  e.  X )  ->  ( -u 1 S D )  e.  X
)
1310, 12anim12dan 845 . . . 4  |-  ( ( W  e.  CVecOLD  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( -u 1 S C )  e.  X  /\  ( -u 1 S D )  e.  X ) )
14133adant2 1024 . . 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 26027 . . 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 1309 . 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 2470 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   ran crn 4855   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   CCcc 9536   1c1 9539   -ucneg 9860   CVecOLDcvc 26009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-grpo 25764  df-ablo 25855  df-vc 26010
This theorem is referenced by: (None)
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