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Theorem th3qlem2 6116
Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
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, ⟩)))
Assertion
Ref Expression
th3qlem2 ((A ((𝑆 × 𝑆) / ) B ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
Distinct variable groups:   z,w,v,u,𝑡,𝑠,f,g,,   z,𝑆,w,v,u,𝑡,𝑠,f,g,   z,A,w,v,u,𝑡,𝑠,f   z,B,w,v,u,𝑡,𝑠,f   z, + ,w,v,u,𝑡,𝑠,f,g,
Allowed substitution hints:   A(g,)   B(g,)

Proof of Theorem th3qlem2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 th3q.2 . . 3 Er (𝑆 × 𝑆)
2 eqid 2018 . . . . 5 (𝑆 × 𝑆) = (𝑆 × 𝑆)
3 breq1 3737 . . . . . . . 8 (⟨w, v⟩ = 𝑠 → (⟨w, vu, 𝑡⟩ ↔ 𝑠 u, 𝑡⟩))
43anbi1d 441 . . . . . . 7 (⟨w, v⟩ = 𝑠 → ((⟨w, vu, 𝑡 x y) ↔ (𝑠 u, 𝑡 x y)))
5 oveq1 5439 . . . . . . . 8 (⟨w, v⟩ = 𝑠 → (⟨w, v+ x) = (𝑠 + x))
65breq1d 3744 . . . . . . 7 (⟨w, v⟩ = 𝑠 → ((⟨w, v+ x) (⟨u, 𝑡+ y) ↔ (𝑠 + x) (⟨u, 𝑡+ y)))
74, 6imbi12d 223 . . . . . 6 (⟨w, v⟩ = 𝑠 → (((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y)) ↔ ((𝑠 u, 𝑡 x y) → (𝑠 + x) (⟨u, 𝑡+ y))))
87imbi2d 219 . . . . 5 (⟨w, v⟩ = 𝑠 → (((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y))) ↔ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((𝑠 u, 𝑡 x y) → (𝑠 + x) (⟨u, 𝑡+ y)))))
9 breq2 3738 . . . . . . . 8 (⟨u, 𝑡⟩ = f → (𝑠 u, 𝑡⟩ ↔ 𝑠 f))
109anbi1d 441 . . . . . . 7 (⟨u, 𝑡⟩ = f → ((𝑠 u, 𝑡 x y) ↔ (𝑠 f x y)))
11 oveq1 5439 . . . . . . . 8 (⟨u, 𝑡⟩ = f → (⟨u, 𝑡+ y) = (f + y))
1211breq2d 3746 . . . . . . 7 (⟨u, 𝑡⟩ = f → ((𝑠 + x) (⟨u, 𝑡+ y) ↔ (𝑠 + x) (f + y)))
1310, 12imbi12d 223 . . . . . 6 (⟨u, 𝑡⟩ = f → (((𝑠 u, 𝑡 x y) → (𝑠 + x) (⟨u, 𝑡+ y)) ↔ ((𝑠 f x y) → (𝑠 + x) (f + y))))
1413imbi2d 219 . . . . 5 (⟨u, 𝑡⟩ = f → (((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((𝑠 u, 𝑡 x y) → (𝑠 + x) (⟨u, 𝑡+ y))) ↔ ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((𝑠 f x y) → (𝑠 + x) (f + y)))))
15 breq1 3737 . . . . . . . . . 10 (⟨𝑠, f⟩ = x → (⟨𝑠, fg, ⟩ ↔ x g, ⟩))
1615anbi2d 440 . . . . . . . . 9 (⟨𝑠, f⟩ = x → ((⟨w, vu, 𝑡𝑠, fg, ⟩) ↔ (⟨w, vu, 𝑡 x g, ⟩)))
17 oveq2 5440 . . . . . . . . . 10 (⟨𝑠, f⟩ = x → (⟨w, v+𝑠, f⟩) = (⟨w, v+ x))
1817breq1d 3744 . . . . . . . . 9 (⟨𝑠, f⟩ = x → ((⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩) ↔ (⟨w, v+ x) (⟨u, 𝑡+g, ⟩)))
1916, 18imbi12d 223 . . . . . . . 8 (⟨𝑠, f⟩ = x → (((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩)) ↔ ((⟨w, vu, 𝑡 x g, ⟩) → (⟨w, v+ x) (⟨u, 𝑡+g, ⟩))))
2019imbi2d 219 . . . . . . 7 (⟨𝑠, f⟩ = x → ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩))) ↔ (((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡 x g, ⟩) → (⟨w, v+ x) (⟨u, 𝑡+g, ⟩)))))
21 breq2 3738 . . . . . . . . . 10 (⟨g, ⟩ = y → (x g, ⟩ ↔ x y))
2221anbi2d 440 . . . . . . . . 9 (⟨g, ⟩ = y → ((⟨w, vu, 𝑡 x g, ⟩) ↔ (⟨w, vu, 𝑡 x y)))
23 oveq2 5440 . . . . . . . . . 10 (⟨g, ⟩ = y → (⟨u, 𝑡+g, ⟩) = (⟨u, 𝑡+ y))
2423breq2d 3746 . . . . . . . . 9 (⟨g, ⟩ = y → ((⟨w, v+ x) (⟨u, 𝑡+g, ⟩) ↔ (⟨w, v+ x) (⟨u, 𝑡+ y)))
2522, 24imbi12d 223 . . . . . . . 8 (⟨g, ⟩ = y → (((⟨w, vu, 𝑡 x g, ⟩) → (⟨w, v+ x) (⟨u, 𝑡+g, ⟩)) ↔ ((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y))))
2625imbi2d 219 . . . . . . 7 (⟨g, ⟩ = y → ((((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡 x g, ⟩) → (⟨w, v+ x) (⟨u, 𝑡+g, ⟩))) ↔ (((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y)))))
27 th3q.4 . . . . . . . 8 ((((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, ⟩)))
2827expcom 109 . . . . . . 7 (((𝑠 𝑆 f 𝑆) (g 𝑆 𝑆)) → (((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡𝑠, fg, ⟩) → (⟨w, v+𝑠, f⟩) (⟨u, 𝑡+g, ⟩))))
292, 20, 26, 282optocl 4340 . . . . . 6 ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → (((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y))))
3029com12 27 . . . . 5 (((w 𝑆 v 𝑆) (u 𝑆 𝑡 𝑆)) → ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((⟨w, vu, 𝑡 x y) → (⟨w, v+ x) (⟨u, 𝑡+ y))))
312, 8, 14, 302optocl 4340 . . . 4 ((𝑠 (𝑆 × 𝑆) f (𝑆 × 𝑆)) → ((x (𝑆 × 𝑆) y (𝑆 × 𝑆)) → ((𝑠 f x y) → (𝑠 + x) (f + y))))
3231imp 115 . . 3 (((𝑠 (𝑆 × 𝑆) f (𝑆 × 𝑆)) (x (𝑆 × 𝑆) y (𝑆 × 𝑆))) → ((𝑠 f x y) → (𝑠 + x) (f + y)))
331, 32th3qlem1 6115 . 2 ((A ((𝑆 × 𝑆) / ) B ((𝑆 × 𝑆) / )) → ∃*z𝑠x((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ))
34 vex 2534 . . . . . . 7 w V
35 vex 2534 . . . . . . 7 v V
3634, 35opex 3936 . . . . . 6 w, v V
37 vex 2534 . . . . . . 7 u V
38 vex 2534 . . . . . . 7 𝑡 V
3937, 38opex 3936 . . . . . 6 u, 𝑡 V
40 eceq1 6048 . . . . . . . . 9 (𝑠 = ⟨w, v⟩ → [𝑠] = [⟨w, v⟩] )
4140eqeq2d 2029 . . . . . . . 8 (𝑠 = ⟨w, v⟩ → (A = [𝑠] A = [⟨w, v⟩] ))
42 eceq1 6048 . . . . . . . . 9 (x = ⟨u, 𝑡⟩ → [x] = [⟨u, 𝑡⟩] )
4342eqeq2d 2029 . . . . . . . 8 (x = ⟨u, 𝑡⟩ → (B = [x] B = [⟨u, 𝑡⟩] ))
4441, 43bi2anan9 526 . . . . . . 7 ((𝑠 = ⟨w, v x = ⟨u, 𝑡⟩) → ((A = [𝑠] B = [x] ) ↔ (A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] )))
45 oveq12 5441 . . . . . . . . 9 ((𝑠 = ⟨w, v x = ⟨u, 𝑡⟩) → (𝑠 + x) = (⟨w, v+u, 𝑡⟩))
4645eceq1d 6049 . . . . . . . 8 ((𝑠 = ⟨w, v x = ⟨u, 𝑡⟩) → [(𝑠 + x)] = [(⟨w, v+u, 𝑡⟩)] )
4746eqeq2d 2029 . . . . . . 7 ((𝑠 = ⟨w, v x = ⟨u, 𝑡⟩) → (z = [(𝑠 + x)] z = [(⟨w, v+u, 𝑡⟩)] ))
4844, 47anbi12d 445 . . . . . 6 ((𝑠 = ⟨w, v x = ⟨u, 𝑡⟩) → (((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ) ↔ ((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] )))
4936, 39, 48spc2ev 2621 . . . . 5 (((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) → 𝑠x((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ))
5049exlimivv 1754 . . . 4 (u𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) → 𝑠x((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ))
5150exlimivv 1754 . . 3 (wvu𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ) → 𝑠x((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ))
5251moimi 1943 . 2 (∃*z𝑠x((A = [𝑠] B = [x] ) z = [(𝑠 + x)] ) → ∃*zwvu𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
5333, 52syl 14 1 ((A ((𝑆 × 𝑆) / ) B ((𝑆 × 𝑆) / )) → ∃*zwvu𝑡((A = [⟨w, v⟩] B = [⟨u, 𝑡⟩] ) z = [(⟨w, v+u, 𝑡⟩)] ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wex 1358   wcel 1370  ∃*wmo 1879  Vcvv 2531  cop 3349   class class class wbr 3734   × cxp 4266  (class class class)co 5432   Er wer 6010  [cec 6011   / cqs 6012
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fv 4833  df-ov 5435  df-er 6013  df-ec 6015  df-qs 6019
This theorem is referenced by:  th3qcor  6117  th3q  6118
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