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Theorem nvaddsub4 30538
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 12359 . . . . . 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 30511 . . . . . 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 7435 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))))
92, 4nvscl 30507 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
101, 9mp3an2 1445 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
112, 4nvscl 30507 . . . . . . 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 30506 . . . 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 30501 . . . . 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 30501 . . . . 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 30523 . . 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 30523 . . . . 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 30523 . . . . 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 7437 . 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 6549  (class class class)co 7419  cc 11138  1c1 11141  -cneg 11477  NrmCVeccnv 30465   +𝑣 cpv 30466  BaseSetcba 30467   ·𝑠OLD cns 30468  𝑣 cnsb 30470
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-ltxr 11285  df-sub 11478  df-neg 11479  df-grpo 30374  df-gid 30375  df-ginv 30376  df-gdiv 30377  df-ablo 30426  df-vc 30440  df-nv 30473  df-va 30476  df-ba 30477  df-sm 30478  df-0v 30479  df-vs 30480  df-nmcv 30481
This theorem is referenced by:  vacn  30575  minvecolem2  30756
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