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Theorem nvaddsub4 22142
Description: Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubadd.1  |-  X  =  ( BaseSet `  U )
nvsubadd.2  |-  G  =  ( +v `  U
)
nvsubadd.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvaddsub4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) M ( C G D ) )  =  ( ( A M C ) G ( B M D ) ) )

Proof of Theorem nvaddsub4
StepHypRef Expression
1 neg1cn 10067 . . . . . 6  |-  -u 1  e.  CC
2 nvsubadd.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
3 nvsubadd.2 . . . . . . 7  |-  G  =  ( +v `  U
)
4 eqid 2436 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
52, 3, 4nvdi 22111 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  C  e.  X  /\  D  e.  X )
)  ->  ( -u 1
( .s OLD `  U
) ( C G D ) )  =  ( ( -u 1
( .s OLD `  U
) C ) G ( -u 1 ( .s OLD `  U
) D ) ) )
61, 5mp3anr1 1276 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( -u 1
( .s OLD `  U
) ( C G D ) )  =  ( ( -u 1
( .s OLD `  U
) C ) G ( -u 1 ( .s OLD `  U
) D ) ) )
763adant2 976 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( -u 1 ( .s OLD `  U ) ( C G D ) )  =  ( ( -u
1 ( .s OLD `  U ) C ) G ( -u 1
( .s OLD `  U
) D ) ) )
87oveq2d 6097 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) G ( -u
1 ( .s OLD `  U ) ( C G D ) ) )  =  ( ( A G B ) G ( ( -u
1 ( .s OLD `  U ) C ) G ( -u 1
( .s OLD `  U
) D ) ) ) )
92, 4nvscl 22107 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 ( .s OLD `  U ) C )  e.  X )
101, 9mp3an2 1267 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 ( .s OLD `  U ) C )  e.  X )
112, 4nvscl 22107 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  D  e.  X )  ->  ( -u 1 ( .s OLD `  U ) D )  e.  X )
121, 11mp3an2 1267 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  D  e.  X )  ->  ( -u 1 ( .s OLD `  U ) D )  e.  X )
1310, 12anim12dan 811 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( ( -u 1 ( .s OLD `  U ) C )  e.  X  /\  ( -u 1 ( .s OLD `  U ) D )  e.  X ) )
14133adant2 976 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( -u 1 ( .s
OLD `  U ) C )  e.  X  /\  ( -u 1 ( .s OLD `  U
) D )  e.  X ) )
152, 3nvadd4 22106 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( ( -u 1
( .s OLD `  U
) C )  e.  X  /\  ( -u
1 ( .s OLD `  U ) D )  e.  X ) )  ->  ( ( A G B ) G ( ( -u 1
( .s OLD `  U
) C ) G ( -u 1 ( .s OLD `  U
) D ) ) )  =  ( ( A G ( -u
1 ( .s OLD `  U ) C ) ) G ( B G ( -u 1
( .s OLD `  U
) D ) ) ) )
1614, 15syld3an3 1229 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) G ( (
-u 1 ( .s
OLD `  U ) C ) G (
-u 1 ( .s
OLD `  U ) D ) ) )  =  ( ( A G ( -u 1
( .s OLD `  U
) C ) ) G ( B G ( -u 1 ( .s OLD `  U
) D ) ) ) )
178, 16eqtrd 2468 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) G ( -u
1 ( .s OLD `  U ) ( C G D ) ) )  =  ( ( A G ( -u
1 ( .s OLD `  U ) C ) ) G ( B G ( -u 1
( .s OLD `  U
) D ) ) ) )
18 simp1 957 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  U  e.  NrmCVec )
192, 3nvgcl 22099 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
20193expb 1154 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  X
)
21203adant3 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( A G B )  e.  X )
222, 3nvgcl 22099 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  C  e.  X  /\  D  e.  X )  ->  ( C G D )  e.  X )
23223expb 1154 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( C G D )  e.  X
)
24233adant2 976 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( C G D )  e.  X )
25 nvsubadd.3 . . . 4  |-  M  =  ( -v `  U
)
262, 3, 4, 25nvmval 22123 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A G B )  e.  X  /\  ( C G D )  e.  X )  ->  (
( A G B ) M ( C G D ) )  =  ( ( A G B ) G ( -u 1 ( .s OLD `  U
) ( C G D ) ) ) )
2718, 21, 24, 26syl3anc 1184 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) M ( C G D ) )  =  ( ( A G B ) G ( -u 1 ( .s OLD `  U
) ( C G D ) ) ) )
282, 3, 4, 25nvmval 22123 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  C  e.  X )  ->  ( A M C )  =  ( A G (
-u 1 ( .s
OLD `  U ) C ) ) )
29283adant3r 1181 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( A M C )  =  ( A G ( -u
1 ( .s OLD `  U ) C ) ) )
30293adant2r 1179 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( A M C )  =  ( A G (
-u 1 ( .s
OLD `  U ) C ) ) )
312, 3, 4, 25nvmval 22123 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  D  e.  X )  ->  ( B M D )  =  ( B G (
-u 1 ( .s
OLD `  U ) D ) ) )
32313adant3l 1180 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( B M D )  =  ( B G ( -u
1 ( .s OLD `  U ) D ) ) )
33323adant2l 1178 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( B M D )  =  ( B G (
-u 1 ( .s
OLD `  U ) D ) ) )
3430, 33oveq12d 6099 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A M C ) G ( B M D ) )  =  ( ( A G ( -u 1
( .s OLD `  U
) C ) ) G ( B G ( -u 1 ( .s OLD `  U
) D ) ) ) )
3517, 27, 343eqtr4d 2478 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  (
( A G B ) M ( C G D ) )  =  ( ( A M C ) G ( B M D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991   -ucneg 9292   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   -vcnsb 22068
This theorem is referenced by:  vacn  22190  minvecolem2  22377
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079
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