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Theorem nvaddsub4 30686
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 = ( +𝑣𝑈)
nvpncan2.3 𝑀 = ( −𝑣𝑈)
Assertion
Ref Expression
nvaddsub4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Proof of Theorem nvaddsub4
StepHypRef Expression
1 neg1cn 12378 . . . . . 6 -1 ∈ ℂ
2 nvpncan2.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
3 nvpncan2.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
4 eqid 2735 . . . . . . 7 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
52, 3, 4nvdi 30659 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
61, 5mp3anr1 1457 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
763adant2 1130 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
87oveq2d 7447 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))))
92, 4nvscl 30655 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
101, 9mp3an2 1448 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
112, 4nvscl 30655 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
121, 11mp3an2 1448 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
1310, 12anim12dan 619 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
14133adant2 1130 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
152, 3nvadd4 30654 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
1614, 15syld3an3 1408 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
178, 16eqtrd 2775 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
18 simp1 1135 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → 𝑈 ∈ NrmCVec)
192, 3nvgcl 30649 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20193expb 1119 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21203adant3 1131 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
222, 3nvgcl 30649 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐷𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23223expb 1119 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24233adant2 1130 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
25 nvpncan2.3 . . . 4 𝑀 = ( −𝑣𝑈)
262, 3, 4, 25nvmval 30671 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
2718, 21, 24, 26syl3anc 1370 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
282, 3, 4, 25nvmval 30671 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
29283adant3r 1180 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
30293adant2r 1178 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
312, 3, 4, 25nvmval 30671 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐷𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
32313adant3l 1179 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
33323adant2l 1177 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
3430, 33oveq12d 7449 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
3517, 27, 343eqtr4d 2785 1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  cc 11151  1c1 11154  -cneg 11491  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616  𝑣 cnsb 30618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298  df-sub 11492  df-neg 11493  df-grpo 30522  df-gid 30523  df-ginv 30524  df-gdiv 30525  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-vs 30628  df-nmcv 30629
This theorem is referenced by:  vacn  30723  minvecolem2  30904
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