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⟩ ∈ V ∧ ⟨C, D⟩ ∈ V))
2		opexb 4604	. . . 4 ⊢ (⟨A, B⟩ ∈ V ↔ (A ∈ V ∧ B ∈ V))
3		opexb 4604	. . . 4 ⊢ (⟨C, D⟩ ∈ V ↔ (C ∈ V ∧ D ∈ V))
4	2, 3	anbi12i 678	. . 3 ⊢ ((⟨A, B⟩ ∈ V ∧ ⟨C, D⟩ ∈ V) ↔ ((A ∈ V ∧ B ∈ V) ∧ (C ∈ V ∧ D ∈ V)))
5	1, 4	sylib 188	. 2 ⊢ (⟨A, B⟩ PProd (R, S)⟨C, D⟩ → ((A ∈ V ∧ B ∈ V) ∧ (C ∈ V ∧ D ∈ V)))
6		brex 4690	. . . 4 ⊢ (ARC → (A ∈ V ∧ C ∈ V))
7		brex 4690	. . . 4 ⊢ (BSD → (B ∈ V ∧ D ∈ V))
8	6, 7	anim12i 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)))
10	8, 9	sylibr 203	. 2 ⊢ ((ARC ∧ BSD) → ((A ∈ V ∧ B ∈ V) ∧ (C ∈ V ∧ D ∈ 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 ∈ 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 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 ∈ 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 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 ∘ 1st ) ⊗ (S ∘ 2nd ))
34	33	breqi 4646	. . . . . . 7 ⊢ (⟨x, y⟩ PProd (R, S)⟨z, w⟩ ↔ ⟨x, y⟩((R ∘ 1st ) ⊗ (S ∘ 2nd ))⟨z, w⟩)
35		trtxp 5782	. . . . . . 7 ⊢ (⟨x, y⟩((R ∘ 1st ) ⊗ (S ∘ 2nd ))⟨z, w⟩ ↔ (⟨x, y⟩(R ∘ 1st )z ∧ ⟨x, y⟩(S ∘ 2nd )w))
36	34, 35	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ PProd (R, S)⟨z, w⟩ ↔ (⟨x, y⟩(R ∘ 1st )z ∧ ⟨x, y⟩(S ∘ 2nd )w))
37		brco 4884	. . . . . . . . 9 ⊢ (⟨x, y⟩(R ∘ 1st )z ↔ ∃a(⟨x, y⟩1st a ∧ aRz))
38		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
39		vex 2863	. . . . . . . . . . . . 13 ⊢ y ∈ V
40	38, 39	opbr1st 5502	. . . . . . . . . . . 12 ⊢ (⟨x, y⟩1st a ↔ x = a)
41		eqcom 2355	. . . . . . . . . . . 12 ⊢ (x = a ↔ a = x)
42	40, 41	bitri 240	. . . . . . . . . . 11 ⊢ (⟨x, y⟩1st a ↔ a = x)
43	42	anbi1i 676	. . . . . . . . . 10 ⊢ ((⟨x, y⟩1st a ∧ aRz) ↔ (a = x ∧ aRz))
44	43	exbii 1582	. . . . . . . . 9 ⊢ (∃a(⟨x, y⟩1st a ∧ aRz) ↔ ∃a(a = x ∧ aRz))
45	37, 44	bitri 240	. . . . . . . 8 ⊢ (⟨x, y⟩(R ∘ 1st )z ↔ ∃a(a = x ∧ aRz))
46		breq1 4643	. . . . . . . . 9 ⊢ (a = x → (aRz ↔ xRz))
47	38, 46	ceqsexv 2895	. . . . . . . 8 ⊢ (∃a(a = x ∧ aRz) ↔ xRz)
48	45, 47	bitri 240	. . . . . . 7 ⊢ (⟨x, y⟩(R ∘ 1st )z ↔ xRz)
49		brco 4884	. . . . . . . . 9 ⊢ (⟨x, y⟩(S ∘ 2nd )w ↔ ∃a(⟨x, y⟩2nd a ∧ aSw))
50	38, 39	opbr2nd 5503	. . . . . . . . . . . 12 ⊢ (⟨x, y⟩2nd a ↔ y = a)
51		eqcom 2355	. . . . . . . . . . . 12 ⊢ (y = a ↔ a = y)
52	50, 51	bitri 240	. . . . . . . . . . 11 ⊢ (⟨x, y⟩2nd a ↔ a = y)
53	52	anbi1i 676	. . . . . . . . . 10 ⊢ ((⟨x, y⟩2nd a ∧ aSw) ↔ (a = y ∧ aSw))
54	53	exbii 1582	. . . . . . . . 9 ⊢ (∃a(⟨x, y⟩2nd a ∧ aSw) ↔ ∃a(a = y ∧ aSw))
55	49, 54	bitri 240	. . . . . . . 8 ⊢ (⟨x, y⟩(S ∘ 2nd )w ↔ ∃a(a = y ∧ aSw))
56		breq1 4643	. . . . . . . . 9 ⊢ (a = y → (aSw ↔ ySw))
57	39, 56	ceqsexv 2895	. . . . . . . 8 ⊢ (∃a(a = y ∧ aSw) ↔ ySw)
58	55, 57	bitri 240	. . . . . . 7 ⊢ (⟨x, y⟩(S ∘ 2nd )w ↔ ySw)
59	48, 58	anbi12i 678	. . . . . 6 ⊢ ((⟨x, y⟩(R ∘ 1st )z ∧ ⟨x, y⟩(S ∘ 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 ∈ V ∧ D ∈ V) → (⟨x, y⟩ PProd (R, S)⟨C, D⟩ ↔ (xRC ∧ ySD)))
62	16, 22, 61	vtocl2g 2919	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ((C ∈ V ∧ D ∈ V) → (⟨A, B⟩ PProd (R, S)⟨C, D⟩ ↔ (ARC ∧ BSD))))
63	62	imp 418	. 2 ⊢ (((A ∈ V ∧ B ∈ V) ∧ (C ∈ V ∧ D ∈ 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  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722  2^nd c2nd 4784   ⊗ 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