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Theorem th3q 6147
 Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
th3q.1 V
th3q.2 Er (𝑆 × 𝑆)
th3q.4 ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))
th3q.5 𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}
Assertion
Ref Expression
th3q (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] )
Distinct variable groups:   x,y,z,w,v,u,𝑡,𝑠,f,g,,   x,𝑆,y,z,w,v,u,𝑡,𝑠,f,g,   x,A,y,z,w,v,u,𝑡,𝑠,f   x,B,y,z,w,v,u,𝑡,𝑠,f   x,𝐶,y,z,w,v,u,𝑡   x,𝐷,y,z,w,v,u,𝑡   x, + ,y,z,w,v,u,𝑡,𝑠,f,g,
Allowed substitution hints:   A(g,)   B(g,)   𝐶(f,g,,𝑠)   𝐷(f,g,,𝑠)   𝐺(x,y,z,w,v,u,𝑡,f,g,,𝑠)

Proof of Theorem th3q
StepHypRef Expression
1 opelxpi 4319 . . . 4 ((A 𝑆 B 𝑆) → ⟨A, B (𝑆 × 𝑆))
2 th3q.1 . . . . 5 V
32ecelqsi 6096 . . . 4 (⟨A, B (𝑆 × 𝑆) → [⟨A, B⟩] ((𝑆 × 𝑆) / ))
41, 3syl 14 . . 3 ((A 𝑆 B 𝑆) → [⟨A, B⟩] ((𝑆 × 𝑆) / ))
5 opelxpi 4319 . . . 4 ((𝐶 𝑆 𝐷 𝑆) → ⟨𝐶, 𝐷 (𝑆 × 𝑆))
62ecelqsi 6096 . . . 4 (⟨𝐶, 𝐷 (𝑆 × 𝑆) → [⟨𝐶, 𝐷⟩] ((𝑆 × 𝑆) / ))
75, 6syl 14 . . 3 ((𝐶 𝑆 𝐷 𝑆) → [⟨𝐶, 𝐷⟩] ((𝑆 × 𝑆) / ))
84, 7anim12i 321 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] ((𝑆 × 𝑆) / ) [⟨𝐶, 𝐷⟩] ((𝑆 × 𝑆) / )))
9 eqid 2037 . . . 4 [⟨A, B⟩] = [⟨A, B⟩]
10 eqid 2037 . . . 4 [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩]
119, 10pm3.2i 257 . . 3 ([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )
12 eqid 2037 . . 3 [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)]
13 opeq12 3542 . . . . . 6 ((w = A v = B) → ⟨w, v⟩ = ⟨A, B⟩)
14 eceq1 6077 . . . . . . . . 9 (⟨w, v⟩ = ⟨A, B⟩ → [⟨w, v⟩] = [⟨A, B⟩] )
1514eqeq2d 2048 . . . . . . . 8 (⟨w, v⟩ = ⟨A, B⟩ → ([⟨A, B⟩] = [⟨w, v⟩] ↔ [⟨A, B⟩] = [⟨A, B⟩] ))
1615anbi1d 438 . . . . . . 7 (⟨w, v⟩ = ⟨A, B⟩ → (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) ↔ ([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
17 oveq1 5462 . . . . . . . . 9 (⟨w, v⟩ = ⟨A, B⟩ → (⟨w, v+𝐶, 𝐷⟩) = (⟨A, B+𝐶, 𝐷⟩))
1817eceq1d 6078 . . . . . . . 8 (⟨w, v⟩ = ⟨A, B⟩ → [(⟨w, v+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] )
1918eqeq2d 2048 . . . . . . 7 (⟨w, v⟩ = ⟨A, B⟩ → ([(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ↔ [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] ))
2016, 19anbi12d 442 . . . . . 6 (⟨w, v⟩ = ⟨A, B⟩ → ((([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ) ↔ (([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] )))
2113, 20syl 14 . . . . 5 ((w = A v = B) → ((([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ) ↔ (([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] )))
2221spc2egv 2636 . . . 4 ((A 𝑆 B 𝑆) → ((([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] ) → wv(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] )))
23 opeq12 3542 . . . . . . 7 ((u = 𝐶 𝑡 = 𝐷) → ⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
24 eceq1 6077 . . . . . . . . . 10 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨u, 𝑡⟩] = [⟨𝐶, 𝐷⟩] )
2524eqeq2d 2048 . . . . . . . . 9 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ))
2625anbi2d 437 . . . . . . . 8 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) ↔ ([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] )))
27 oveq2 5463 . . . . . . . . . 10 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (⟨w, v+u, 𝑡⟩) = (⟨w, v+𝐶, 𝐷⟩))
2827eceq1d 6078 . . . . . . . . 9 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [(⟨w, v+u, 𝑡⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] )
2928eqeq2d 2048 . . . . . . . 8 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ↔ [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ))
3026, 29anbi12d 442 . . . . . . 7 (⟨u, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ((([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ) ↔ (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] )))
3123, 30syl 14 . . . . . 6 ((u = 𝐶 𝑡 = 𝐷) → ((([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ) ↔ (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] )))
3231spc2egv 2636 . . . . 5 ((𝐶 𝑆 𝐷 𝑆) → ((([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ) → u𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
33322eximdv 1759 . . . 4 ((𝐶 𝑆 𝐷 𝑆) → (wv(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+𝐶, 𝐷⟩)] ) → wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
3422, 33sylan9 389 . . 3 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ((([⟨A, B⟩] = [⟨A, B⟩] [⟨𝐶, 𝐷⟩] = [⟨𝐶, 𝐷⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨A, B+𝐶, 𝐷⟩)] ) → wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
3511, 12, 34mp2ani 408 . 2 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ))
36 ecexg 6046 . . . 4 ( V → [(⟨A, B+𝐶, 𝐷⟩)] V)
372, 36ax-mp 7 . . 3 [(⟨A, B+𝐶, 𝐷⟩)] V
38 eqeq1 2043 . . . . . . . 8 (x = [⟨A, B⟩] → (x = [⟨w, v⟩] ↔ [⟨A, B⟩] = [⟨w, v⟩] ))
39 eqeq1 2043 . . . . . . . 8 (y = [⟨𝐶, 𝐷⟩] → (y = [⟨u, 𝑡⟩] ↔ [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ))
4038, 39bi2anan9 538 . . . . . . 7 ((x = [⟨A, B⟩] y = [⟨𝐶, 𝐷⟩] ) → ((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) ↔ ([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] )))
41 eqeq1 2043 . . . . . . 7 (z = [(⟨A, B+𝐶, 𝐷⟩)] → (z = [(⟨w, v+u, 𝑡⟩)] ↔ [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ))
4240, 41bi2anan9 538 . . . . . 6 (((x = [⟨A, B⟩] y = [⟨𝐶, 𝐷⟩] ) z = [(⟨A, B+𝐶, 𝐷⟩)] ) → (((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) ↔ (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
43423impa 1098 . . . . 5 ((x = [⟨A, B⟩] y = [⟨𝐶, 𝐷⟩] z = [(⟨A, B+𝐶, 𝐷⟩)] ) → (((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) ↔ (([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
44434exbidv 1747 . . . 4 ((x = [⟨A, B⟩] y = [⟨𝐶, 𝐷⟩] z = [(⟨A, B+𝐶, 𝐷⟩)] ) → (wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) ↔ wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] )))
45 th3q.2 . . . . 5 Er (𝑆 × 𝑆)
46 th3q.4 . . . . 5 ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) ((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆))) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)))
472, 45, 46th3qlem2 6145 . . . 4 ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
48 th3q.5 . . . 4 𝐺 = {⟨⟨x, y⟩, z⟩ ∣ ((x ((𝑆 × 𝑆) / ) y ((𝑆 × 𝑆) / )) wvu𝑡((x = [⟨w, v⟩] y = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))}
4944, 47, 48ovig 5564 . . 3 (([⟨A, B⟩] ((𝑆 × 𝑆) / ) [⟨𝐶, 𝐷⟩] ((𝑆 × 𝑆) / ) [(⟨A, B+𝐶, 𝐷⟩)] V) → (wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ) → ([⟨A, B⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] ))
5037, 49mp3an3 1220 . 2 (([⟨A, B⟩] ((𝑆 × 𝑆) / ) [⟨𝐶, 𝐷⟩] ((𝑆 × 𝑆) / )) → (wvu𝑡(([⟨A, B⟩] = [⟨w, v⟩] [⟨𝐶, 𝐷⟩] = [⟨u, 𝑡⟩] ) [(⟨A, B+𝐶, 𝐷⟩)] = [(⟨w, v+u, 𝑡⟩)] ) → ([⟨A, B⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] ))
518, 35, 50sylc 56 1 (((A 𝑆 B 𝑆) (𝐶 𝑆 𝐷 𝑆)) → ([⟨A, B⟩] 𝐺[⟨𝐶, 𝐷⟩] ) = [(⟨A, B+𝐶, 𝐷⟩)] )
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   class class class wbr 3755   × cxp 4286  (class class class)co 5455  {coprab 5456   Er wer 6039  [cec 6040   / cqs 6041 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-er 6042  df-ec 6044  df-qs 6048 This theorem is referenced by:  oviec  6148
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