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Theorem enq0enq 7203
 Description: Equivalence on positive fractions in terms of equivalence on nonnegative 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 7196 . . 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 4513 . . 3 ((N × N) × (N × N)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))}
31, 2ineq12i 3243 . 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 4639 . 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 534 . . . . . 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 558 . . . . . . . 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 7081 . . . . . . . . . . . . 13 (𝑥N𝑥 ∈ ω)
87ssriv 3069 . . . . . . . . . . . 12 N ⊆ ω
9 xpss1 4617 . . . . . . . . . . . 12 (N ⊆ ω → (N × N) ⊆ (ω × N))
108, 9ax-mp 5 . . . . . . . . . . 11 (N × N) ⊆ (ω × N)
1110sseli 3061 . . . . . . . . . 10 (𝑥 ∈ (N × N) → 𝑥 ∈ (ω × N))
1211pm4.71ri 387 . . . . . . . . 9 (𝑥 ∈ (N × N) ↔ (𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)))
1310sseli 3061 . . . . . . . . . 10 (𝑦 ∈ (N × N) → 𝑦 ∈ (ω × N))
1413pm4.71ri 387 . . . . . . . . 9 (𝑦 ∈ (N × N) ↔ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N)))
1512, 14anbi12i 453 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)) ∧ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N))))
166, 15bitr4i 186 . . . . . . 7 (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))
1716anbi1i 451 . . . . . 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 183 . . . . 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 2178 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝑧, 𝑤⟩ ∈ (N × N)))
20 opelxp 4537 . . . . . . . . . . . . . . . . . . 19 (⟨𝑧, 𝑤⟩ ∈ (N × N) ↔ (𝑧N𝑤N))
2119, 20syl6bb 195 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ (𝑧N𝑤N)))
22 eleq1 2178 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝑣, 𝑢⟩ ∈ (N × N)))
23 opelxp 4537 . . . . . . . . . . . . . . . . . . 19 (⟨𝑣, 𝑢⟩ ∈ (N × N) ↔ (𝑣N𝑢N))
2422, 23syl6bb 195 . . . . . . . . . . . . . . . . . 18 (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ (𝑣N𝑢N)))
2521, 24bi2anan9 578 . . . . . . . . . . . . . . . . 17 ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))))
2625pm5.32i 447 . . . . . . . . . . . . . . . 16 (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧N𝑤N) ∧ (𝑣N𝑢N))))
2726anbi1i 451 . . . . . . . . . . . . . . 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 396 . . . . . . . . . . . . . . 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 183 . . . . . . . . . . . . . 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 7089 . . . . . . . . . . . . . . . . . 18 ((𝑧N𝑢N) → (𝑧 ·N 𝑢) = (𝑧 ·o 𝑢))
31 mulpiord 7089 . . . . . . . . . . . . . . . . . 18 ((𝑤N𝑣N) → (𝑤 ·N 𝑣) = (𝑤 ·o 𝑣))
3230, 31eqeqan12d 2131 . . . . . . . . . . . . . . . . 17 (((𝑧N𝑢N) ∧ (𝑤N𝑣N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
3332an42s 561 . . . . . . . . . . . . . . . 16 (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
3433pm5.32i 447 . . . . . . . . . . . . . . 15 ((((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ (((𝑧N𝑤N) ∧ (𝑣N𝑢N)) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
3534anbi2i 450 . . . . . . . . . . . . . 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 186 . . . . . . . . . . . . 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 396 . . . . . . . . . . . . 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 186 . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 186 . . . . . . . . . . 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 264 . . . . . . . . . . . 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 451 . . . . . . . . . . 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 451 . . . . . . . . . . 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 209 . . . . . . . . . 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 396 . . . . . . . . . 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 396 . . . . . . . . . 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 209 . . . . . . . . 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 1568 . . . . . . . 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 1863 . . . . . . . 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 1863 . . . . . . . 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 209 . . . . . . 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 1568 . . . . . 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 1863 . . . . . 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 1863 . . . . . 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 209 . . . . 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 183 . . . 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 3963 . . 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 7119 . . 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 2139 . 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 2141 1 ~Q = ( ~Q0 ∩ ((N × N) × (N × N)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1314  ∃wex 1451   ∈ wcel 1463   ∩ cin 3038   ⊆ wss 3039  ⟨cop 3498  {copab 3956  ωcom 4472   × cxp 4505  (class class class)co 5740   ·o comu 6277  Ncnpi 7044   ·N cmi 7046   ~Q ceq 7051   ~Q0 ceq0 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-res 4519  df-iota 5056  df-fv 5099  df-ov 5743  df-ni 7076  df-mi 7078  df-enq 7119  df-enq0 7196 This theorem is referenced by:  nqnq0pi  7210
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