Theorem qrpprod 5837

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.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 |- (<.A, B>. PProd (R, S)<.C, D>. -> (<.A, B>. e. _V /\ <.C, D>. e. _V))
2		opexb 4604	. . . 4 |- (<.A, B>. e. _V <-> (A e. _V /\ B e. _V))
3		opexb 4604	. . . 4 |- (<.C, D>. e. _V <-> (C e. _V /\ D e. _V))
4	2, 3	anbi12i 678	. . 3 |- ((<.A, B>. e. _V /\ <.C, D>. e. _V) <-> ((A e. _V /\ B e. _V) /\ (C e. _V /\ D e. _V)))
5	1, 4	sylib 188	. 2 |- (<.A, B>. PProd (R, S)<.C, D>. -> ((A e. _V /\ B e. _V) /\ (C e. _V /\ D e. _V)))
6		brex 4690	. . . 4 |- (ARC -> (A e. _V /\ C e. _V))
7		brex 4690	. . . 4 |- (BSD -> (B e. _V /\ D e. _V))
8	6, 7	anim12i 549	. . 3 |- ((ARC /\ BSD) -> ((A e. _V /\ C e. _V) /\ (B e. _V /\ D e. _V)))
9		an4 797	. . 3 |- (((A e. _V /\ B e. _V) /\ (C e. _V /\ D e. _V)) <-> ((A e. _V /\ C e. _V) /\ (B e. _V /\ D e. _V)))
10	8, 9	sylibr 203	. 2 |- ((ARC /\ BSD) -> ((A e. _V /\ B e. _V) /\ (C e. _V /\ D e. _V)))
11		opeq1 4579	. . . . . . 7 |- (x = A -> <.x, y>. = <.A, y>.)
12	11	breq1d 4650	. . . . . 6 |- (x = A -> (<.x, y>. PProd (R, S)<.C, D>. <-> <.A, y>. PProd (R, S)<.C, D>.))
13		breq1 4643	. . . . . . 7 |- (x = A -> (xRC <-> ARC))
14	13	anbi1d 685	. . . . . 6 |- (x = A -> ((xRC /\ ySD) <-> (ARC /\ ySD)))
15	12, 14	bibi12d 312	. . . . 5 |- (x = A -> ((<.x, y>. PProd (R, S)<.C, D>. <-> (xRC /\ ySD)) <-> (<.A, y>. PProd (R, S)<.C, D>. <-> (ARC /\ ySD))))
16	15	imbi2d 307	. . . 4 |- (x = A -> (((C e. _V /\ D e. _V) -> (<.x, y>. PProd (R, S)<.C, D>. <-> (xRC /\ ySD))) <-> ((C e. _V /\ D e. _V) -> (<.A, y>. PProd (R, S)<.C, D>. <-> (ARC /\ ySD)))))
17		opeq2 4580	. . . . . . 7 |- (y = B -> <.A, y>. = <.A, B>.)
18	17	breq1d 4650	. . . . . 6 |- (y = B -> (<.A, y>. PProd (R, S)<.C, D>. <-> <.A, B>. PProd (R, S)<.C, D>.))
19		breq1 4643	. . . . . . 7 |- (y = B -> (ySD <-> BSD))
20	19	anbi2d 684	. . . . . 6 |- (y = B -> ((ARC /\ ySD) <-> (ARC /\ BSD)))
21	18, 20	bibi12d 312	. . . . 5 |- (y = B -> ((<.A, y>. PProd (R, S)<.C, D>. <-> (ARC /\ ySD)) <-> (<.A, B>. PProd (R, S)<.C, D>. <-> (ARC /\ BSD))))
22	21	imbi2d 307	. . . 4 |- (y = B -> (((C e. _V /\ D e. _V) -> (<.A, y>. PProd (R, S)<.C, D>. <-> (ARC /\ ySD))) <-> ((C e. _V /\ D e. _V) -> (<.A, B>. PProd (R, S)<.C, D>. <-> (ARC /\ BSD)))))
23		opeq1 4579	. . . . . . 7 |- (z = C -> <.z, w>. = <.C, w>.)
24	23	breq2d 4652	. . . . . 6 |- (z = C -> (<.x, y>. PProd (R, S)<.z, w>. <-> <.x, y>. PProd (R, S)<.C, w>.))
25		breq2 4644	. . . . . . 7 |- (z = C -> (xRz <-> xRC))
26	25	anbi1d 685	. . . . . 6 |- (z = C -> ((xRz /\ ySw) <-> (xRC /\ ySw)))
27	24, 26	bibi12d 312	. . . . 5 |- (z = C -> ((<.x, y>. PProd (R, S)<.z, w>. <-> (xRz /\ ySw)) <-> (<.x, y>. PProd (R, S)<.C, w>. <-> (xRC /\ ySw))))
28		opeq2 4580	. . . . . . 7 |- (w = D -> <.C, w>. = <.C, D>.)
29	28	breq2d 4652	. . . . . 6 |- (w = D -> (<.x, y>. PProd (R, S)<.C, w>. <-> <.x, y>. PProd (R, S)<.C, D>.))
30		breq2 4644	. . . . . . 7 |- (w = D -> (ySw <-> ySD))
31	30	anbi2d 684	. . . . . 6 |- (w = D -> ((xRC /\ ySw) <-> (xRC /\ ySD)))
32	29, 31	bibi12d 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 5739	. . . . . . . 8 |- PProd (R, S) = ((R o. 1st) (x) (S o. 2nd))
34	33	breqi 4646	. . . . . . 7 |- (<.x, y>. PProd (R, S)<.z, w>. <-> <.x, y>.((R o. 1st) (x) (S o. 2nd))<.z, w>.)
35		trtxp 5782	. . . . . . 7 |- (<.x, y>.((R o. 1st) (x) (S o. 2nd))<.z, w>. <-> (<.x, y>.(R o. 1st)z /\ <.x, y>.(S o. 2nd)w))
36	34, 35	bitri 240	. . . . . 6 |- (<.x, y>. PProd (R, S)<.z, w>. <-> (<.x, y>.(R o. 1st)z /\ <.x, y>.(S o. 2nd)w))
37		brco 4884	. . . . . . . . 9 |- (<.x, y>.(R o. 1st)z <-> E.a(<.x, y>.1sta /\ aRz))
38		vex 2863	. . . . . . . . . . . . 13 |- x e. _V
39		vex 2863	. . . . . . . . . . . . 13 |- y e. _V
40	38, 39	opbr1st 5502	. . . . . . . . . . . 12 |- (<.x, y>.1sta <-> x = a)
41		eqcom 2355	. . . . . . . . . . . 12 |- (x = a <-> a = x)
42	40, 41	bitri 240	. . . . . . . . . . 11 |- (<.x, y>.1sta <-> a = x)
43	42	anbi1i 676	. . . . . . . . . 10 |- ((<.x, y>.1sta /\ aRz) <-> (a = x /\ aRz))
44	43	exbii 1582	. . . . . . . . 9 |- (E.a(<.x, y>.1sta /\ aRz) <-> E.a(a = x /\ aRz))
45	37, 44	bitri 240	. . . . . . . 8 |- (<.x, y>.(R o. 1st)z <-> E.a(a = x /\ aRz))
46		breq1 4643	. . . . . . . . 9 |- (a = x -> (aRz <-> xRz))
47	38, 46	ceqsexv 2895	. . . . . . . 8 |- (E.a(a = x /\ aRz) <-> xRz)
48	45, 47	bitri 240	. . . . . . 7 |- (<.x, y>.(R o. 1st)z <-> xRz)
49		brco 4884	. . . . . . . . 9 |- (<.x, y>.(S o. 2nd)w <-> E.a(<.x, y>.2nda /\ aSw))
50	38, 39	opbr2nd 5503	. . . . . . . . . . . 12 |- (<.x, y>.2nda <-> y = a)
51		eqcom 2355	. . . . . . . . . . . 12 |- (y = a <-> a = y)
52	50, 51	bitri 240	. . . . . . . . . . 11 |- (<.x, y>.2nda <-> a = y)
53	52	anbi1i 676	. . . . . . . . . 10 |- ((<.x, y>.2nda /\ aSw) <-> (a = y /\ aSw))
54	53	exbii 1582	. . . . . . . . 9 |- (E.a(<.x, y>.2nda /\ aSw) <-> E.a(a = y /\ aSw))
55	49, 54	bitri 240	. . . . . . . 8 |- (<.x, y>.(S o. 2nd)w <-> E.a(a = y /\ aSw))
56		breq1 4643	. . . . . . . . 9 |- (a = y -> (aSw <-> ySw))
57	39, 56	ceqsexv 2895	. . . . . . . 8 |- (E.a(a = y /\ aSw) <-> ySw)
58	55, 57	bitri 240	. . . . . . 7 |- (<.x, y>.(S o. 2nd)w <-> ySw)
59	48, 58	anbi12i 678	. . . . . 6 |- ((<.x, y>.(R o. 1st)z /\ <.x, y>.(S o. 2nd)w) <-> (xRz /\ ySw))
60	36, 59	bitri 240	. . . . 5 |- (<.x, y>. PProd (R, S)<.z, w>. <-> (xRz /\ ySw))
61	27, 32, 60	vtocl2g 2919	. . . 4 |- ((C e. _V /\ D e. _V) -> (<.x, y>. PProd (R, S)<.C, D>. <-> (xRC /\ ySD)))
62	16, 22, 61	vtocl2g 2919	. . 3 |- ((A e. _V /\ B e. _V) -> ((C e. _V /\ D e. _V) -> (<.A, B>. PProd (R, S)<.C, D>. <-> (ARC /\ BSD))))
63	62	imp 418	. 2 |- (((A e. _V /\ B e. _V) /\ (C e. _V /\ D e. _V)) -> (<.A, B>. PProd (R, S)<.C, D>. <-> (ARC /\ BSD)))
64	5, 10, 63	pm5.21nii 342	1 |- (<.A, B>. PProd (R, S)<.C, D>. <-> (ARC /\ BSD))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  <.cop 4562   class class class wbr 4640  1stc1st 4718   o. ccom 4722  2ndc2nd 4784   (x) ctxp 5736   PProd cpprod 5738

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-cnv 4786  df-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  dmfrec  6317  fnfreclem2  6319  fnfreclem3  6320  frecsuc  6323