Theorem nvaddsub4 30748

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 12139	. . . . . 6 ⊢ -1 ∈ ℂ
2		nvpncan2.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nvpncan2.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		eqid 2741	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
5	2, 3, 4	nvdi 30721	. . . . . 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 1467	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
7	6	3adant2 1138	. . . 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 7375	. . 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 30717	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
10	1, 9	mp3an2 1458	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
11	2, 4	nvscl 30717	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
12	1, 11	mp3an2 1458	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
13	10, 12	anim12dan 626	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋))
14	13	3adant2 1138	. . . 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 30716	. . . 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 1418	. . 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 2776	. 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 1143	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
19	2, 3	nvgcl 30711	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20	19	3expb 1127	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21	20	3adant3 1139	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
22	2, 3	nvgcl 30711	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23	22	3expb 1127	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24	23	3adant2 1138	. . 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 30733	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))))
27	18, 21, 24, 26	syl3anc 1380	. 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 30733	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
29	28	3adant3r 1189	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
30	29	3adant2r 1187	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
31	2, 3, 4, 25	nvmval 30733	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
32	31	3adant3l 1188	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
33	32	3adant2l 1186	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
34	30, 33	oveq12d 7377	. 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 2786	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ‘cfv 6488  (class class class)co 7359  ℂcc 11032  1c1 11035  -cneg 11374  NrmCVeccnv 30675   +_𝑣 cpv 30676  BaseSetcba 30677   ·_𝑠OLD cns 30678   −_𝑣 cnsb 30680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-po 5528  df-so 5529  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11177  df-mnf 11178  df-ltxr 11180  df-sub 11375  df-neg 11376  df-grpo 30584  df-gid 30585  df-ginv 30586  df-gdiv 30587  df-ablo 30636  df-vc 30650  df-nv 30683  df-va 30686  df-ba 30687  df-sm 30688  df-0v 30689  df-vs 30690  df-nmcv 30691

This theorem is referenced by:  vacn  30785  minvecolem2  30966