New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  qrpprod GIF version

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
 Copyright terms: Public domain W3C validator