Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvaddsub4 Structured version   Visualization version   GIF version

 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)))

StepHypRef Expression
1 neg1cn 11479 . . . . . 6 -1 ∈ ℂ
2 nvpncan2.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
3 nvpncan2.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
4 eqid 2825 . . . . . . 7 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
52, 3, 4nvdi 28036 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
61, 5mp3anr1 1586 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
763adant2 1165 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
87oveq2d 6926 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))))
92, 4nvscl 28032 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
101, 9mp3an2 1577 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
112, 4nvscl 28032 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
121, 11mp3an2 1577 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
1310, 12anim12dan 612 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
14133adant2 1165 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
152, 3nvadd4 28031 . . . 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 1532 . . 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 2861 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
18 simp1 1170 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → 𝑈 ∈ NrmCVec)
192, 3nvgcl 28026 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20193expb 1153 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21203adant3 1166 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
222, 3nvgcl 28026 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐷𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23223expb 1153 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24233adant2 1165 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
25 nvpncan2.3 . . . 4 𝑀 = ( −𝑣𝑈)
262, 3, 4, 25nvmval 28048 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
2718, 21, 24, 26syl3anc 1494 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
282, 3, 4, 25nvmval 28048 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
29283adant3r 1235 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
30293adant2r 1231 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
312, 3, 4, 25nvmval 28048 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐷𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
32313adant3l 1233 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
33323adant2l 1229 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
3430, 33oveq12d 6928 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
3517, 27, 343eqtr4d 2871 1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ‘cfv 6127  (class class class)co 6910  ℂcc 10257  1c1 10260  -cneg 10593  NrmCVeccnv 27990   +𝑣 cpv 27991  BaseSetcba 27992   ·𝑠OLD cns 27993   −𝑣 cnsb 27995 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-ltxr 10403  df-sub 10594  df-neg 10595  df-grpo 27899  df-gid 27900  df-ginv 27901  df-gdiv 27902  df-ablo 27951  df-vc 27965  df-nv 27998  df-va 28001  df-ba 28002  df-sm 28003  df-0v 28004  df-vs 28005  df-nmcv 28006 This theorem is referenced by:  vacn  28100  minvecolem2  28282
 Copyright terms: Public domain W3C validator