Theorem nvaddsub4 30636

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 12147	. . . . . 6 ⊢ -1 ∈ ℂ
2		nvpncan2.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nvpncan2.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		eqid 2729	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
5	2, 3, 4	nvdi 30609	. . . . . 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 1460	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
7	6	3adant2 1131	. . . 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 7385	. . 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 30605	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
10	1, 9	mp3an2 1451	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
11	2, 4	nvscl 30605	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
12	1, 11	mp3an2 1451	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
13	10, 12	anim12dan 619	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋))
14	13	3adant2 1131	. . . 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 30604	. . . 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 1411	. . 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 2764	. 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 1136	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
19	2, 3	nvgcl 30599	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20	19	3expb 1120	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21	20	3adant3 1132	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
22	2, 3	nvgcl 30599	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23	22	3expb 1120	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24	23	3adant2 1131	. . 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 30621	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))))
27	18, 21, 24, 26	syl3anc 1373	. 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 30621	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
29	28	3adant3r 1182	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
30	29	3adant2r 1180	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
31	2, 3, 4, 25	nvmval 30621	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
32	31	3adant3l 1181	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
33	32	3adant2l 1179	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
34	30, 33	oveq12d 7387	. 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 2774	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  1c1 11045  -cneg 11382  NrmCVeccnv 30563   +_𝑣 cpv 30564  BaseSetcba 30565   ·_𝑠OLD cns 30566   −_𝑣 cnsb 30568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-ltxr 11189  df-sub 11383  df-neg 11384  df-grpo 30472  df-gid 30473  df-ginv 30474  df-gdiv 30475  df-ablo 30524  df-vc 30538  df-nv 30571  df-va 30574  df-ba 30575  df-sm 30576  df-0v 30577  df-vs 30578  df-nmcv 30579

This theorem is referenced by:  vacn  30673  minvecolem2  30854