Theorem nvmdi 27503

Description: Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvmdi.1	|- X = ( BaseSet ` U )
nvmdi.3	|- M = ( -v ` U )
nvmdi.4	|- S = ( .sOLD ` U )

Assertion

Ref	Expression
nvmdi	|- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B M C ) ) = ( ( A S B ) M ( A S C ) ) )

Proof of Theorem nvmdi

Step	Hyp	Ref	Expression
1		simpr1 1067	. . . . 5 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> A e. CC )
2		simpr2 1068	. . . . 5 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> B e. X )
3		neg1cn 11124	. . . . . . 7 |- -u 1 e. CC
4		nvmdi.1	. . . . . . . 8 |- X = ( BaseSet ` U )
5		nvmdi.4	. . . . . . . 8 |- S = ( .sOLD ` U )
6	4, 5	nvscl 27481	. . . . . . 7 |- ( ( U e. NrmCVec /\ -u 1 e. CC /\ C e. X ) -> ( -u 1 S C ) e. X )
7	3, 6	mp3an2 1412	. . . . . 6 |- ( ( U e. NrmCVec /\ C e. X ) -> ( -u 1 S C ) e. X )
8	7	3ad2antr3 1228	. . . . 5 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( -u 1 S C ) e. X )
9	1, 2, 8	3jca 1242	. . . 4 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A e. CC /\ B e. X /\ ( -u 1 S C ) e. X ) )
10		eqid 2622	. . . . 5 |- ( +v ` U ) = ( +v ` U )
11	4, 10, 5	nvdi 27485	. . . 4 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ ( -u 1 S C ) e. X ) ) -> ( A S ( B ( +v ` U ) ( -u 1 S C ) ) ) = ( ( A S B ) ( +v ` U ) ( A S ( -u 1 S C ) ) ) )
12	9, 11	syldan 487	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B ( +v ` U ) ( -u 1 S C ) ) ) = ( ( A S B ) ( +v ` U ) ( A S ( -u 1 S C ) ) ) )
13	4, 5	nvscom 27484	. . . . . 6 |- ( ( U e. NrmCVec /\ ( A e. CC /\ -u 1 e. CC /\ C e. X ) ) -> ( A S ( -u 1 S C ) ) = ( -u 1 S ( A S C ) ) )
14	3, 13	mp3anr2 1422	. . . . 5 |- ( ( U e. NrmCVec /\ ( A e. CC /\ C e. X ) ) -> ( A S ( -u 1 S C ) ) = ( -u 1 S ( A S C ) ) )
15	14	3adantr2 1221	. . . 4 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( -u 1 S C ) ) = ( -u 1 S ( A S C ) ) )
16	15	oveq2d 6666	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) ( +v ` U ) ( A S ( -u 1 S C ) ) ) = ( ( A S B ) ( +v ` U ) ( -u 1 S ( A S C ) ) ) )
17	12, 16	eqtrd 2656	. 2 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B ( +v ` U ) ( -u 1 S C ) ) ) = ( ( A S B ) ( +v ` U ) ( -u 1 S ( A S C ) ) ) )
18		nvmdi.3	. . . . 5 |- M = ( -v ` U )
19	4, 10, 5, 18	nvmval 27497	. . . 4 |- ( ( U e. NrmCVec /\ B e. X /\ C e. X ) -> ( B M C ) = ( B ( +v ` U ) ( -u 1 S C ) ) )
20	19	3adant3r1 1274	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( B M C ) = ( B ( +v ` U ) ( -u 1 S C ) ) )
21	20	oveq2d 6666	. 2 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B M C ) ) = ( A S ( B ( +v ` U ) ( -u 1 S C ) ) ) )
22		simpl 473	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> U e. NrmCVec )
23	4, 5	nvscl 27481	. . . 4 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( A S B ) e. X )
24	23	3adant3r3 1276	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S B ) e. X )
25	4, 5	nvscl 27481	. . . 4 |- ( ( U e. NrmCVec /\ A e. CC /\ C e. X ) -> ( A S C ) e. X )
26	25	3adant3r2 1275	. . 3 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S C ) e. X )
27	4, 10, 5, 18	nvmval 27497	. . 3 |- ( ( U e. NrmCVec /\ ( A S B ) e. X /\ ( A S C ) e. X ) -> ( ( A S B ) M ( A S C ) ) = ( ( A S B ) ( +v ` U ) ( -u 1 S ( A S C ) ) ) )
28	22, 24, 26, 27	syl3anc 1326	. 2 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( ( A S B ) M ( A S C ) ) = ( ( A S B ) ( +v ` U ) ( -u 1 S ( A S C ) ) ) )
29	17, 21, 28	3eqtr4d 2666	1 |- ( ( U e. NrmCVec /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( A S ( B M C ) ) = ( ( A S B ) M ( A S C ) ) )

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:  smcnlem  27552  minvecolem2  27731