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Theorem nvaddsub4 28426
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 11743 . . . . . 6 -1 ∈ ℂ
2 nvpncan2.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
3 nvpncan2.2 . . . . . . 7 𝐺 = ( +𝑣𝑈)
4 eqid 2819 . . . . . . 7 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
52, 3, 4nvdi 28399 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ 𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
61, 5mp3anr1 1451 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
763adant2 1125 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷)) = ((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷)))
87oveq2d 7164 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺𝐵)𝐺((-1( ·𝑠OLD𝑈)𝐶)𝐺(-1( ·𝑠OLD𝑈)𝐷))))
92, 4nvscl 28395 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
101, 9mp3an2 1442 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋) → (-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋)
112, 4nvscl 28395 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
121, 11mp3an2 1442 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝐷𝑋) → (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋)
1310, 12anim12dan 620 . . . . 5 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
14133adant2 1125 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((-1( ·𝑠OLD𝑈)𝐶) ∈ 𝑋 ∧ (-1( ·𝑠OLD𝑈)𝐷) ∈ 𝑋))
152, 3nvadd4 28394 . . . 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 1403 . . 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 2854 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
18 simp1 1130 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → 𝑈 ∈ NrmCVec)
192, 3nvgcl 28389 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
20193expb 1114 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
21203adant3 1126 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝐺𝐵) ∈ 𝑋)
222, 3nvgcl 28389 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐷𝑋) → (𝐶𝐺𝐷) ∈ 𝑋)
23223expb 1114 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
24233adant2 1125 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐶𝐺𝐷) ∈ 𝑋)
25 nvpncan2.3 . . . 4 𝑀 = ( −𝑣𝑈)
262, 3, 4, 25nvmval 28411 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐶𝐺𝐷) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
2718, 21, 24, 26syl3anc 1365 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝐺𝐵)𝐺(-1( ·𝑠OLD𝑈)(𝐶𝐺𝐷))))
282, 3, 4, 25nvmval 28411 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
29283adant3r 1175 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
30293adant2r 1173 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐴𝑀𝐶) = (𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶)))
312, 3, 4, 25nvmval 28411 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐷𝑋) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
32313adant3l 1174 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐵𝑋 ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
33323adant2l 1172 . . 3 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → (𝐵𝑀𝐷) = (𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷)))
3430, 33oveq12d 7166 . 2 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)) = ((𝐴𝐺(-1( ·𝑠OLD𝑈)𝐶))𝐺(𝐵𝐺(-1( ·𝑠OLD𝑈)𝐷))))
3517, 27, 343eqtr4d 2864 1 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  cc 10527  1c1 10530  -cneg 10863  NrmCVeccnv 28353   +𝑣 cpv 28354  BaseSetcba 28355   ·𝑠OLD cns 28356  𝑣 cnsb 28358
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-ltxr 10672  df-sub 10864  df-neg 10865  df-grpo 28262  df-gid 28263  df-ginv 28264  df-gdiv 28265  df-ablo 28314  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-vs 28368  df-nmcv 28369
This theorem is referenced by:  vacn  28463  minvecolem2  28644
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