Theorem nvmdi 8255

Description: Distributive law for scalar product over subtraction.

Hypotheses

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

Assertion

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

Proof of Theorem nvmdi

Step	Hyp	Ref	Expression
1		3simp1 787	. . . . . 6 |- ((A e. CC /\ B e. X /\ C e. X) -> A e. CC)
2	1	adantl 388	. . . . 5 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> A e. CC)
3		3simp2 788	. . . . . 6 |- ((A e. CC /\ B e. X /\ C e. X) -> B e. X)
4	3	adantl 388	. . . . 5 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> B e. X)
5		ax1cn 5256	. . . . . . . 8 |- 1 e. CC
6	5	negcl 5356	. . . . . . 7 |- -u1 e. CC
7		nvmdi.1	. . . . . . . 8 |- X = (Base` U)
8		nvmdi.4	. . . . . . . 8 |- S = (.s` U)
9	7, 8	nvscl 8232	. . . . . . 7 |- ((U e. NrmCVec /\ -u1 e. CC /\ C e. X) -> (-u1SC) e. X)
10	6, 9	mp3an2 903	. . . . . 6 |- ((U e. NrmCVec /\ C e. X) -> (-u1SC) e. X)
11	10	3ad2antr3 813	. . . . 5 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (-u1SC) e. X)
12	2, 4, 11	3jca 818	. . . 4 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (A e. CC /\ B e. X /\ (-u1SC) e. X))
13		eqid 1475	. . . . 5 |- (+v` U) = (+v` U)
14	7, 13, 8	nvdi 8236	. . . 4 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ (-u1SC) e. X)) -> (AS(B(+v` U)(-u1SC))) = ((ASB)(+v` U)(AS(-u1SC))))
15	12, 14	syldan 467	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(B(+v` U)(-u1SC))) = ((ASB)(+v` U)(AS(-u1SC))))
16	7, 8	nvscom 8235	. . . . . 6 |- ((U e. NrmCVec /\ (A e. CC /\ -u1 e. CC /\ C e. X)) -> (AS(-u1SC)) = (-u1S(ASC)))
17	6, 16	mp3anr2 913	. . . . 5 |- ((U e. NrmCVec /\ (A e. CC /\ C e. X)) -> (AS(-u1SC)) = (-u1S(ASC)))
18	17	3adantr2 806	. . . 4 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(-u1SC)) = (-u1S(ASC)))
19	18	opreq2d 3973	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> ((ASB)(+v` U)(AS(-u1SC))) = ((ASB)(+v` U)(-u1S(ASC))))
20	15, 19	eqtrd 1506	. 2 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(B(+v` U)(-u1SC))) = ((ASB)(+v` U)(-u1S(ASC))))
21		nvmdi.3	. . . . 5 |- M = (-v` U)
22	7, 13, 8, 21	nvmval 8248	. . . 4 |- ((U e. NrmCVec /\ B e. X /\ C e. X) -> (BMC) = (B(+v` U)(-u1SC)))
23	22	3adant3r1 841	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (BMC) = (B(+v` U)(-u1SC)))
24	23	opreq2d 3973	. 2 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(BMC)) = (AS(B(+v` U)(-u1SC))))
25	7, 13, 8, 21	nvmval 8248	. . 3 |- ((U e. NrmCVec /\ (ASB) e. X /\ (ASC) e. X) -> ((ASB)M(ASC)) = ((ASB)(+v` U)(-u1S(ASC))))
26		pm3.26 319	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> U e. NrmCVec)
27	7, 8	nvscl 8232	. . . 4 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (ASB) e. X)
28	27	3adant3r3 843	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (ASB) e. X)
29	7, 8	nvscl 8232	. . . 4 |- ((U e. NrmCVec /\ A e. CC /\ C e. X) -> (ASC) e. X)
30	29	3adant3r2 842	. . 3 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (ASC) e. X)
31	25, 26, 28, 30	syl3anc 857	. 2 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> ((ASB)M(ASC)) = ((ASB)(+v` U)(-u1S(ASC))))
32	20, 24, 31	3eqtr4d 1516	1 |- ((U e. NrmCVec /\ (A e. CC /\ B e. X /\ C e. X)) -> (AS(BMC)) = ((ASB)M(ASC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  ` cfv 3179  (class class class)co 3960  CCcc 5219  1c1 5222  -ucneg 5280  NrmCVeccnv 8188  +vcpv 8189  Basecba 8190  .scns 8191  -vcnsb 8193

This theorem is referenced by:  minveclem19 8547  minveclem35 8563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-sub 5343  df-neg 5345  df-grp 8020  df-gid 8021  df-ginv 8022  df-gdiv 8023  df-abl 8084  df-vc 8150  df-nv 8196  df-va 8199  df-ba 8200  df-sm 8201  df-0v 8202  df-vs 8203  df-nm 8204

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