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Theorem nvaddsub4 27512
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
nvpncan2.1 |- X = ( BaseSet ` U )
nvpncan2.2 |- G = ( +v ` U )
nvpncan2.3 |- 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 11124 . . . . . 6 |- -u 1 e. CC
2 nvpncan2.1 . . . . . . 7 |- X = ( BaseSet ` U )
3 nvpncan2.2 . . . . . . 7 |- G = ( +v ` U )
4 eqid 2622 . . . . . . 7 |- ( .sOLD ` U ) = ( .sOLD ` U )
52, 3, 4nvdi 27485 . . . . . 6 |- ( ( U e. NrmCVec /\ ( -u 1 e. CC /\ C e. X /\ D e. X ) ) -> ( -u 1 ( .sOLD ` U ) ( C G D ) ) = ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) )
61, 5mp3anr1 1421 . . . . 5 |- ( ( U e. NrmCVec /\ ( C e. X /\ D e. X ) ) -> ( -u 1 ( .sOLD ` U ) ( C G D ) ) = ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) )
763adant2 1080 . . . 4 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( -u 1 ( .sOLD ` U ) ( C G D ) ) = ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) )
87oveq2d 6666 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( -u 1 ( .sOLD ` U ) ( C G D ) ) ) = ( ( A G B ) G ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) ) )
92, 4nvscl 27481 . . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ C e. X ) -> ( -u 1 ( .sOLD ` U ) C ) e. X )
101, 9mp3an2 1412 . . . . . 6 |- ( ( U e. NrmCVec /\ C e. X ) -> ( -u 1 ( .sOLD ` U ) C ) e. X )
112, 4nvscl 27481 . . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ D e. X ) -> ( -u 1 ( .sOLD ` U ) D ) e. X )
121, 11mp3an2 1412 . . . . . 6 |- ( ( U e. NrmCVec /\ D e. X ) -> ( -u 1 ( .sOLD ` U ) D ) e. X )
1310, 12anim12dan 882 . . . . 5 |- ( ( U e. NrmCVec /\ ( C e. X /\ D e. X ) ) -> ( ( -u 1 ( .sOLD ` U ) C ) e. X /\ ( -u 1 ( .sOLD ` U ) D ) e. X ) )
14133adant2 1080 . . . 4 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( -u 1 ( .sOLD ` U ) C ) e. X /\ ( -u 1 ( .sOLD ` U ) D ) e. X ) )
152, 3nvadd4 27480 . . . 4 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( ( -u 1 ( .sOLD ` U ) C ) e. X /\ ( -u 1 ( .sOLD ` U ) D ) e. X ) ) -> ( ( A G B ) G ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) ) = ( ( A G ( -u 1 ( .sOLD ` U ) C ) ) G ( B G ( -u 1 ( .sOLD ` U ) D ) ) ) )
1614, 15syld3an3 1371 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( ( -u 1 ( .sOLD ` U ) C ) G ( -u 1 ( .sOLD ` U ) D ) ) ) = ( ( A G ( -u 1 ( .sOLD ` U ) C ) ) G ( B G ( -u 1 ( .sOLD ` U ) D ) ) ) )
178, 16eqtrd 2656 . 2 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( -u 1 ( .sOLD ` U ) ( C G D ) ) ) = ( ( A G ( -u 1 ( .sOLD ` U ) C ) ) G ( B G ( -u 1 ( .sOLD ` U ) D ) ) ) )
18 simp1 1061 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> U e. NrmCVec )
192, 3nvgcl 27475 . . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
20193expb 1266 . . . 4 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) ) -> ( A G B ) e. X )
21203adant3 1081 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( A G B ) e. X )
222, 3nvgcl 27475 . . . . 5 |- ( ( U e. NrmCVec /\ C e. X /\ D e. X ) -> ( C G D ) e. X )
23223expb 1266 . . . 4 |- ( ( U e. NrmCVec /\ ( C e. X /\ D e. X ) ) -> ( C G D ) e. X )
24233adant2 1080 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( C G D ) e. X )
25 nvpncan2.3 . . . 4 |- M = ( -v ` U )
262, 3, 4, 25nvmval 27497 . . 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 ( .sOLD ` U ) ( C G D ) ) ) )
2718, 21, 24, 26syl3anc 1326 . 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 ( .sOLD ` U ) ( C G D ) ) ) )
282, 3, 4, 25nvmval 27497 . . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ C e. X ) -> ( A M C ) = ( A G ( -u 1 ( .sOLD ` U ) C ) ) )
29283adant3r 1323 . . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ ( C e. X /\ D e. X ) ) -> ( A M C ) = ( A G ( -u 1 ( .sOLD ` U ) C ) ) )
30293adant2r 1321 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( A M C ) = ( A G ( -u 1 ( .sOLD ` U ) C ) ) )
312, 3, 4, 25nvmval 27497 . . . . 5 |- ( ( U e. NrmCVec /\ B e. X /\ D e. X ) -> ( B M D ) = ( B G ( -u 1 ( .sOLD ` U ) D ) ) )
32313adant3l 1322 . . . 4 |- ( ( U e. NrmCVec /\ B e. X /\ ( C e. X /\ D e. X ) ) -> ( B M D ) = ( B G ( -u 1 ( .sOLD ` U ) D ) ) )
33323adant2l 1320 . . 3 |- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( B M D ) = ( B G ( -u 1 ( .sOLD ` U ) D ) ) )
3430, 33oveq12d 6668 . 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 ( .sOLD ` U ) C ) ) G ( B G ( -u 1 ( .sOLD ` U ) D ) ) ) )
3517, 27, 343eqtr4d 2666 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 setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   -ucneg 10267   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   -vcnsb 27444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455
This theorem is referenced by:  vacn  27549  minvecolem2  27731
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