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Theorem qrpprod 5836
Description: A quadratic relationship over a parallel product. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
qrpprod (A, B PProd (R, S)C, D ↔ (ARC BSD))

Proof of Theorem qrpprod
Dummy variables w a x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A, B PProd (R, S)C, D → (A, B V C, D V))
2 opexb 4603 . . . 4 (A, B V ↔ (A V B V))
3 opexb 4603 . . . 4 (C, D V ↔ (C V D V))
42, 3anbi12i 678 . . 3 ((A, B V C, D V) ↔ ((A V B V) (C V D V)))
51, 4sylib 188 . 2 (A, B PProd (R, S)C, D → ((A V B V) (C V D V)))
6 brex 4689 . . . 4 (ARC → (A V C V))
7 brex 4689 . . . 4 (BSD → (B V D V))
86, 7anim12i 549 . . 3 ((ARC BSD) → ((A V C V) (B V D V)))
9 an4 797 . . 3 (((A V B V) (C V D V)) ↔ ((A V C V) (B V D V)))
108, 9sylibr 203 . 2 ((ARC BSD) → ((A V B V) (C V D V)))
11 opeq1 4578 . . . . . . 7 (x = Ax, y = A, y)
1211breq1d 4649 . . . . . 6 (x = A → (x, y PProd (R, S)C, DA, y PProd (R, S)C, D))
13 breq1 4642 . . . . . . 7 (x = A → (xRCARC))
1413anbi1d 685 . . . . . 6 (x = A → ((xRC ySD) ↔ (ARC ySD)))
1512, 14bibi12d 312 . . . . 5 (x = A → ((x, y PProd (R, S)C, D ↔ (xRC ySD)) ↔ (A, y PProd (R, S)C, D ↔ (ARC ySD))))
1615imbi2d 307 . . . 4 (x = A → (((C V D V) → (x, y PProd (R, S)C, D ↔ (xRC ySD))) ↔ ((C V D V) → (A, y PProd (R, S)C, D ↔ (ARC ySD)))))
17 opeq2 4579 . . . . . . 7 (y = BA, y = A, B)
1817breq1d 4649 . . . . . 6 (y = B → (A, y PProd (R, S)C, DA, B PProd (R, S)C, D))
19 breq1 4642 . . . . . . 7 (y = B → (ySDBSD))
2019anbi2d 684 . . . . . 6 (y = B → ((ARC ySD) ↔ (ARC BSD)))
2118, 20bibi12d 312 . . . . 5 (y = B → ((A, y PProd (R, S)C, D ↔ (ARC ySD)) ↔ (A, B PProd (R, S)C, D ↔ (ARC BSD))))
2221imbi2d 307 . . . 4 (y = B → (((C V D V) → (A, y PProd (R, S)C, D ↔ (ARC ySD))) ↔ ((C V D V) → (A, B PProd (R, S)C, D ↔ (ARC BSD)))))
23 opeq1 4578 . . . . . . 7 (z = Cz, w = C, w)
2423breq2d 4651 . . . . . 6 (z = C → (x, y PProd (R, S)z, wx, y PProd (R, S)C, w))
25 breq2 4643 . . . . . . 7 (z = C → (xRzxRC))
2625anbi1d 685 . . . . . 6 (z = C → ((xRz ySw) ↔ (xRC ySw)))
2724, 26bibi12d 312 . . . . 5 (z = C → ((x, y PProd (R, S)z, w ↔ (xRz ySw)) ↔ (x, y PProd (R, S)C, w ↔ (xRC ySw))))
28 opeq2 4579 . . . . . . 7 (w = DC, w = C, D)
2928breq2d 4651 . . . . . 6 (w = D → (x, y PProd (R, S)C, wx, y PProd (R, S)C, D))
30 breq2 4643 . . . . . . 7 (w = D → (ySwySD))
3130anbi2d 684 . . . . . 6 (w = D → ((xRC ySw) ↔ (xRC ySD)))
3229, 31bibi12d 312 . . . . 5 (w = D → ((x, y PProd (R, S)C, w ↔ (xRC ySw)) ↔ (x, y PProd (R, S)C, D ↔ (xRC ySD))))
33 df-pprod 5738 . . . . . . . 8 PProd (R, S) = ((R 1st ) ⊗ (S 2nd ))
3433breqi 4645 . . . . . . 7 (x, y PProd (R, S)z, wx, y((R 1st ) ⊗ (S 2nd ))z, w)
35 trtxp 5781 . . . . . . 7 (x, y((R 1st ) ⊗ (S 2nd ))z, w ↔ (x, y(R 1st )z x, y(S 2nd )w))
3634, 35bitri 240 . . . . . 6 (x, y PProd (R, S)z, w ↔ (x, y(R 1st )z x, y(S 2nd )w))
37 brco 4883 . . . . . . . . 9 (x, y(R 1st )za(x, y1st a aRz))
38 vex 2862 . . . . . . . . . . . . 13 x V
39 vex 2862 . . . . . . . . . . . . 13 y V
4038, 39opbr1st 5501 . . . . . . . . . . . 12 (x, y1st ax = a)
41 eqcom 2355 . . . . . . . . . . . 12 (x = aa = x)
4240, 41bitri 240 . . . . . . . . . . 11 (x, y1st aa = x)
4342anbi1i 676 . . . . . . . . . 10 ((x, y1st a aRz) ↔ (a = x aRz))
4443exbii 1582 . . . . . . . . 9 (a(x, y1st a aRz) ↔ a(a = x aRz))
4537, 44bitri 240 . . . . . . . 8 (x, y(R 1st )za(a = x aRz))
46 breq1 4642 . . . . . . . . 9 (a = x → (aRzxRz))
4738, 46ceqsexv 2894 . . . . . . . 8 (a(a = x aRz) ↔ xRz)
4845, 47bitri 240 . . . . . . 7 (x, y(R 1st )zxRz)
49 brco 4883 . . . . . . . . 9 (x, y(S 2nd )wa(x, y2nd a aSw))
5038, 39opbr2nd 5502 . . . . . . . . . . . 12 (x, y2nd ay = a)
51 eqcom 2355 . . . . . . . . . . . 12 (y = aa = y)
5250, 51bitri 240 . . . . . . . . . . 11 (x, y2nd aa = y)
5352anbi1i 676 . . . . . . . . . 10 ((x, y2nd a aSw) ↔ (a = y aSw))
5453exbii 1582 . . . . . . . . 9 (a(x, y2nd a aSw) ↔ a(a = y aSw))
5549, 54bitri 240 . . . . . . . 8 (x, y(S 2nd )wa(a = y aSw))
56 breq1 4642 . . . . . . . . 9 (a = y → (aSwySw))
5739, 56ceqsexv 2894 . . . . . . . 8 (a(a = y aSw) ↔ ySw)
5855, 57bitri 240 . . . . . . 7 (x, y(S 2nd )wySw)
5948, 58anbi12i 678 . . . . . 6 ((x, y(R 1st )z x, y(S 2nd )w) ↔ (xRz ySw))
6036, 59bitri 240 . . . . 5 (x, y PProd (R, S)z, w ↔ (xRz ySw))
6127, 32, 60vtocl2g 2918 . . . 4 ((C V D V) → (x, y PProd (R, S)C, D ↔ (xRC ySD)))
6216, 22, 61vtocl2g 2918 . . 3 ((A V B V) → ((C V D V) → (A, B PProd (R, S)C, D ↔ (ARC BSD))))
6362imp 418 . 2 (((A V B V) (C V D V)) → (A, B PProd (R, S)C, D ↔ (ARC BSD)))
645, 10, 63pm5.21nii 342 1 (A, B PProd (R, S)C, D ↔ (ARC BSD))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cop 4561   class class class wbr 4639  1st c1st 4717   ccom 4721  2nd c2nd 4783  ctxp 5735   PProd cpprod 5737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-cnv 4785  df-2nd 4797  df-txp 5736  df-pprod 5738
This theorem is referenced by:  dmfrec  6316  fnfreclem2  6318  fnfreclem3  6319  frecsuc  6322
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