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Theorem enq0enq 6937
Description: Equivalence on positive fractions in terms of equivalence on non-negative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
Assertion
Ref Expression
enq0enq ~Q = ( ~Q0 ∩ ((N × N) × (N × N)))

Proof of Theorem enq0enq
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enq0 6930 . . 3 ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))}
2 df-xp 4419 . . 3 ((N × N) × (N × N)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))}
31, 2ineq12i 3188 . 2 ( ~Q0 ∩ ((N × N) × (N × N))) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))})
4 inopab 4538 . 2 ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))}) = {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))}
5 an32 527 . . . . . 6 ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
6 an4 551 . . . . . . . 8 (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)) ∧ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N))))
7 pinn 6815 . . . . . . . . . . . . 13 (𝑥N𝑥 ∈ ω)
87ssriv 3018 . . . . . . . . . . . 12 N ⊆ ω
9 xpss1 4518 . . . . . . . . . . . 12 (N ⊆ ω → (N × N) ⊆ (ω × N))
108, 9ax-mp 7 . . . . . . . . . . 11 (N × N) ⊆ (ω × N)
1110sseli 3010 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 ∈ (ω × N))
1211pm4.71ri 384 . . . . . . . . 9 (𝑥 ∈ (N × N) ↔ (𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)))
1310sseli 3010 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 ∈ (ω × N))
1413pm4.71ri 384 . . . . . . . . 9 (𝑦 ∈ (N × N) ↔ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N)))
1512, 14anbi12i 448 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)) ∧ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N))))
166, 15bitr4i 185 . . . . . . 7 (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))
1716anbi1i 446 . . . . . 6 ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
185, 17bitri 182 . . . . 5 ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
19 eleq1 2147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝑧, 𝑤⟩ ∈ (N × N)))
20 opelxp 4442 . . . . . . . . . . . . . . . . . . 19 (⟨𝑧, 𝑤⟩ ∈ (N × N) ↔ (𝑧N𝑤N))
2119, 20syl6bb 194 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ (𝑧N𝑤N)))
22 eleq1 2147 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝑣, 𝑢⟩ ∈ (N × N)))
23 opelxp 4442 . . . . . . . . . . . . . . . . . . 19 (⟨𝑣, 𝑢⟩ ∈ (N × N) ↔ (𝑣N𝑢N))
2422, 23syl6bb 194 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ (𝑣N𝑢N)))
2521, 24bi2anan9 571 . . . . . . . . . . . . . . . . 17 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))))
2625pm5.32i 442 . . . . . . . . . . . . . . . 16 (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))))
2726anbi1i 446 . . . . . . . . . . . . . . 15 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
28 anass 393 . . . . . . . . . . . . . . 15 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
2927, 28bitri 182 . . . . . . . . . . . . . 14 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
30 mulpiord 6823 . . . . . . . . . . . . . . . . . 18 ((𝑧N𝑢N) → (𝑧 ·N 𝑢) = (𝑧 ·𝑜 𝑢))
31 mulpiord 6823 . . . . . . . . . . . . . . . . . 18 ((𝑤N𝑣N) → (𝑤 ·N 𝑣) = (𝑤 ·𝑜 𝑣))
3230, 31eqeqan12d 2100 . . . . . . . . . . . . . . . . 17 (((𝑧N𝑢N) ∧ (𝑤N𝑣N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
3332an42s 554 . . . . . . . . . . . . . . . 16 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
3433pm5.32i 442 . . . . . . . . . . . . . . 15 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
3534anbi2i 445 . . . . . . . . . . . . . 14 (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
3629, 35bitr4i 185 . . . . . . . . . . . . 13 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
37 anass 393 . . . . . . . . . . . . 13 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
3836, 37bitr4i 185 . . . . . . . . . . . 12 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
3926anbi1i 446 . . . . . . . . . . . 12 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
4038, 39bitr4i 185 . . . . . . . . . . 11 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
41 ancom 262 . . . . . . . . . . . 12 (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
4241anbi1i 446 . . . . . . . . . . 11 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
4341anbi1i 446 . . . . . . . . . . 11 ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
4440, 42, 433bitr3i 208 . . . . . . . . . 10 ((((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
45 anass 393 . . . . . . . . . 10 ((((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
46 anass 393 . . . . . . . . . 10 ((((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
4744, 45, 463bitr3i 208 . . . . . . . . 9 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
48472exbii 1540 . . . . . . . 8 (∃𝑣𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ∃𝑣𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
49 19.42vv 1833 . . . . . . . 8 (∃𝑣𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
50 19.42vv 1833 . . . . . . . 8 (∃𝑣𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
5148, 49, 503bitr3i 208 . . . . . . 7 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
52512exbii 1540 . . . . . 6 (∃𝑧𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ∃𝑧𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
53 19.42vv 1833 . . . . . 6 (∃𝑧𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
54 19.42vv 1833 . . . . . 6 (∃𝑧𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
5552, 53, 543bitr3i 208 . . . . 5 (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
5618, 55bitri 182 . . . 4 ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
5756opabbii 3882 . . 3 {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
58 df-enq 6853 . . 3 ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
5957, 58eqtr4i 2108 . 2 {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))} = ~Q
603, 4, 593eqtrri 2110 1 ~Q = ( ~Q0 ∩ ((N × N) × (N × N)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  cin 2987  wss 2988  cop 3434  {copab 3875  ωcom 4380   × cxp 4411  (class class class)co 5615   ·𝑜 comu 6135  Ncnpi 6778   ·N cmi 6780   ~Q ceq 6785   ~Q0 ceq0 6792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-res 4425  df-iota 4948  df-fv 4991  df-ov 5618  df-ni 6810  df-mi 6812  df-enq 6853  df-enq0 6930
This theorem is referenced by:  nqnq0pi  6944
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