Theorem nvaddsub4 30737

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 12135	. . . . . 6 ⊢ -1 ∈ ℂ
2		nvpncan2.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nvpncan2.2	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		eqid 2737	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
5	2, 3, 4	nvdi 30710	. . . . . 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 1461	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD ‘𝑈)𝐶)𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
7	6	3adant2 1132	. . . 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 7377	. . 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 30706	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
10	1, 9	mp3an2 1452	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋)
11	2, 4	nvscl 30706	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
12	1, 11	mp3an2 1452	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐷 ∈ 𝑋) → (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋)
13	10, 12	anim12dan 620	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((-1( ·𝑠OLD ‘𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD ‘𝑈)𝐷) ∈ 𝑋))
14	13	3adant2 1132	. . . 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 30705	. . . 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 1412	. . 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 2772	. 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 1137	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝑈 ∈ NrmCVec)
19	2, 3	nvgcl 30700	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20	19	3expb 1121	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21	20	3adant3 1133	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
22	2, 3	nvgcl 30700	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23	22	3expb 1121	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24	23	3adant2 1132	. . 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 30722	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD ‘𝑈)(𝐶𝐺𝐷))))
27	18, 21, 24, 26	syl3anc 1374	. 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 30722	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
29	28	3adant3r 1183	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
30	29	3adant2r 1181	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD ‘𝑈)𝐶)))
31	2, 3, 4, 25	nvmval 30722	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
32	31	3adant3l 1182	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
33	32	3adant2l 1180	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD ‘𝑈)𝐷)))
34	30, 33	oveq12d 7379	. 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 2782	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6493  (class class class)co 7361  ℂcc 11029  1c1 11032  -cneg 11370  NrmCVeccnv 30664   +_𝑣 cpv 30665  BaseSetcba 30666   ·_𝑠OLD cns 30667   −_𝑣 cnsb 30669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11173  df-mnf 11174  df-ltxr 11176  df-sub 11371  df-neg 11372  df-grpo 30573  df-gid 30574  df-ginv 30575  df-gdiv 30576  df-ablo 30625  df-vc 30639  df-nv 30672  df-va 30675  df-ba 30676  df-sm 30677  df-0v 30678  df-vs 30679  df-nmcv 30680

This theorem is referenced by:  vacn  30774  minvecolem2  30955