Theorem nvaddsub4 26896

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 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Proof of Theorem nvaddsub4

Step	Hyp	Ref	Expression
1		neg1cn 11001	. . . . . 6 ⊢ -1 ∈ ℂ
2		nvpncan2.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nvpncan2.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		eqid 2610	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
5	2, 3, 4	nvdi 26869	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
6	1, 5	mp3anr1 1413	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
7	6	3adant2 1073	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
8	7	oveq2d 6565	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))))
9	2, 4	nvscl 26865	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
10	1, 9	mp3an2 1404	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
11	2, 4	nvscl 26865	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
12	1, 11	mp3an2 1404	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
13	10, 12	anim12dan 878	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋))
14	13	3adant2 1073	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋))
15	2, 3	nvadd4 26864	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))))
16	14, 15	syld3an3 1363	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))))
17	8, 16	eqtrd 2644	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))))
18		simp1 1054	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
19	2, 3	nvgcl 26859	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20	19	3expb 1258	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21	20	3adant3 1074	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
22	2, 3	nvgcl 26859	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23	22	3expb 1258	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24	23	3adant2 1073	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
25		nvpncan2.3	. . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
26	2, 3, 4, 25	nvmval 26881	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))))
27	18, 21, 24, 26	syl3anc 1318	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))))
28	2, 3, 4, 25	nvmval 26881	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
29	28	3adant3r 1315	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
30	29	3adant2r 1313	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
31	2, 3, 4, 25	nvmval 26881	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
32	31	3adant3l 1314	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
33	32	3adant2l 1312	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
34	30, 33	oveq12d 6567	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)) = ((𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷))))
35	17, 27, 34	3eqtr4d 2654	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  1c1 9816  -cneg 10146  NrmCVeccnv 26823   +_𝑣 cpv 26824  BaseSetcba 26825   ·_𝑠OLD cns 26826   −_𝑣 cnsb 26828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-sub 10147  df-neg 10148  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839

This theorem is referenced by:  vacn  26933  minvecolem2  27115