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Theorem nvaddsub4 21219
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 9813 . . . . . 6  |-  -u 1  e.  CC
2 nvsubadd.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
3 nvsubadd.2 . . . . . . 7  |-  G  =  ( +v `  U
)
4 eqid 2283 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
52, 3, 4nvdi 21188 . . . . . 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 1274 . . . . 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 974 . . . 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 5874 . . 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 21184 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  C  e.  X )  ->  ( -u 1 ( .s OLD `  U ) C )  e.  X )
101, 9mp3an2 1265 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  C  e.  X )  ->  ( -u 1 ( .s OLD `  U ) C )  e.  X )
112, 4nvscl 21184 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  D  e.  X )  ->  ( -u 1 ( .s OLD `  U ) D )  e.  X )
121, 11mp3an2 1265 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  D  e.  X )  ->  ( -u 1 ( .s OLD `  U ) D )  e.  X )
1310, 12anim12dan 810 . . . . 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 974 . . . 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 21183 . . . 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 1227 . . 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 2315 . 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 955 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  U  e.  NrmCVec )
192, 3nvgcl 21176 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
20193expb 1152 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G B )  e.  X
)
21203adant3 975 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( A G B )  e.  X )
222, 3nvgcl 21176 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  C  e.  X  /\  D  e.  X )  ->  ( C G D )  e.  X )
23223expb 1152 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( C  e.  X  /\  D  e.  X )
)  ->  ( C G D )  e.  X
)
24233adant2 974 . . 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 21200 . . 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 1182 . 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 21200 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  C  e.  X )  ->  ( A M C )  =  ( A G (
-u 1 ( .s
OLD `  U ) C ) ) )
29283adant3r 1179 . . . 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 1177 . . 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 21200 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  D  e.  X )  ->  ( B M D )  =  ( B G (
-u 1 ( .s
OLD `  U ) D ) ) )
32313adant3l 1178 . . . 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 1176 . . 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 5876 . 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 2325 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   -vcnsb 21145
This theorem is referenced by:  vacn  21267  minvecolem2  21454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156
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