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Theorem nvaddsub4 30530
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 12356 . . . . . 6 -1 ∈ ℂ
2 nvpncan2.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
3 nvpncan2.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
4 eqid 2725 . . . . . . 7 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
52, 3, 4nvdi 30503 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
61, 5mp3anr1 1454 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
763adant2 1128 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
87oveq2d 7433 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))))
92, 4nvscl 30499 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
101, 9mp3an2 1445 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
112, 4nvscl 30499 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
121, 11mp3an2 1445 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
1310, 12anim12dan 617 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
14133adant2 1128 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
152, 3nvadd4 30498 . . . 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 1406 . . 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 2765 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
18 simp1 1133 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → 𝑈 ∈ NrmCVec)
192, 3nvgcl 30493 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20193expb 1117 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21203adant3 1129 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
222, 3nvgcl 30493 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐷𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23223expb 1117 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24233adant2 1128 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
25 nvpncan2.3 . . . 4 𝑀 = ( −𝑣𝑈)
262, 3, 4, 25nvmval 30515 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
2718, 21, 24, 26syl3anc 1368 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
282, 3, 4, 25nvmval 30515 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
29283adant3r 1178 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
30293adant2r 1176 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
312, 3, 4, 25nvmval 30515 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐷𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
32313adant3l 1177 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
33323adant2l 1175 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
3430, 33oveq12d 7435 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
3517, 27, 343eqtr4d 2775 1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  cfv 6547  (class class class)co 7417  cc 11136  1c1 11139  -cneg 11475  NrmCVeccnv 30457   +𝑣 cpv 30458  BaseSetcba 30459   ·𝑠OLD cns 30460  𝑣 cnsb 30462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7992  df-2nd 7993  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-ltxr 11283  df-sub 11476  df-neg 11477  df-grpo 30366  df-gid 30367  df-ginv 30368  df-gdiv 30369  df-ablo 30418  df-vc 30432  df-nv 30465  df-va 30468  df-ba 30469  df-sm 30470  df-0v 30471  df-vs 30472  df-nmcv 30473
This theorem is referenced by:  vacn  30567  minvecolem2  30748
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