Theorem enq0enq 7421

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.

Step	Hyp	Ref	Expression
1		df-enq0 7414	. . 3 ⊢ ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))}
2		df-xp 4629	. . 3 ⊢ ((N × N) × (N × N)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))}
3	1, 2	ineq12i 3334	. 2 ⊢ ( ~Q0 ∩ ((N × N) × (N × N))) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))})
4		inopab 4755	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))}
5		an32 562	. . . . . 6 ⊢ ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((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))) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
6		an4 586	. . . . . . . 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 7299	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ N → 𝑥 ∈ ω)
8	7	ssriv 3159	. . . . . . . . . . . 12 ⊢ N ⊆ ω
9		xpss1 4733	. . . . . . . . . . . 12 ⊢ (N ⊆ ω → (N × N) ⊆ (ω × N))
10	8, 9	ax-mp 5	. . . . . . . . . . 11 ⊢ (N × N) ⊆ (ω × N)
11	10	sseli 3151	. . . . . . . . . 10 ⊢ (𝑥 ∈ (N × N) → 𝑥 ∈ (ω × N))
12	11	pm4.71ri 392	. . . . . . . . 9 ⊢ (𝑥 ∈ (N × N) ↔ (𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)))
13	10	sseli 3151	. . . . . . . . . 10 ⊢ (𝑦 ∈ (N × N) → 𝑦 ∈ (ω × N))
14	13	pm4.71ri 392	. . . . . . . . 9 ⊢ (𝑦 ∈ (N × N) ↔ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N)))
15	12, 14	anbi12i 460	. . . . . . . 8 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑥 ∈ (ω × N) ∧ 𝑥 ∈ (N × N)) ∧ (𝑦 ∈ (ω × N) ∧ 𝑦 ∈ (N × N))))
16	6, 15	bitr4i 187	. . . . . . 7 ⊢ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))
17	16	anbi1i 458	. . . . . 6 ⊢ ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
18	5, 17	bitri 184	. . . . 5 ⊢ ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
19		eleq1 2240	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝑧, 𝑤⟩ ∈ (N × N)))
20		opelxp 4653	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑧, 𝑤⟩ ∈ (N × N) ↔ (𝑧 ∈ N ∧ 𝑤 ∈ N))
21	19, 20	bitrdi 196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑤⟩ → (𝑥 ∈ (N × N) ↔ (𝑧 ∈ N ∧ 𝑤 ∈ N)))
22		eleq1 2240	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝑣, 𝑢⟩ ∈ (N × N)))
23		opelxp 4653	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨𝑣, 𝑢⟩ ∈ (N × N) ↔ (𝑣 ∈ N ∧ 𝑢 ∈ N))
24	22, 23	bitrdi 196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ⟨𝑣, 𝑢⟩ → (𝑦 ∈ (N × N) ↔ (𝑣 ∈ N ∧ 𝑢 ∈ N)))
25	21, 24	bi2anan9 606	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))))
26	25	pm5.32i 454	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))))
27	26	anbi1i 458	. . . . . . . . . . . . . . 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⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
28		anass 401	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
29	27, 28	bitri 184	. . . . . . . . . . . . . 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⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
30		mulpiord 7307	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ N ∧ 𝑢 ∈ N) → (𝑧 ·N 𝑢) = (𝑧 ·o 𝑢))
31		mulpiord 7307	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → (𝑤 ·N 𝑣) = (𝑤 ·o 𝑣))
32	30, 31	eqeqan12d 2193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ N ∧ 𝑢 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
33	32	an42s 589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑧 ·N 𝑢) = (𝑤 ·N 𝑣) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
34	33	pm5.32i 454	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
35	34	anbi2i 457	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
36	29, 35	bitr4i 187	. . . . . . . . . . . . 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⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
37		anass 401	. . . . . . . . . . . . 13 ⊢ ((((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)) ↔ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
38	36, 37	bitr4i 187	. . . . . . . . . . . 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⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
39	26	anbi1i 458	. . . . . . . . . . . 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⟩) ∧ ((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N))) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
40	38, 39	bitr4i 187	. . . . . . . . . . 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 266	. . . . . . . . . . . 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⟩)))
42	41	anbi1i 458	. . . . . . . . . . 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)))
43	41	anbi1i 458	. . . . . . . . . . 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)))
44	40, 42, 43	3bitr3i 210	. . . . . . . . . 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 401	. . . . . . . . . 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 401	. . . . . . . . . 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))))
47	44, 45, 46	3bitr3i 210	. . . . . . . . 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))))
48	47	2exbii 1606	. . . . . . . 8 ⊢ (∃𝑣∃𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ∃𝑣∃𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
49		19.42vv 1911	. . . . . . . 8 ⊢ (∃𝑣∃𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
50		19.42vv 1911	. . . . . . . 8 ⊢ (∃𝑣∃𝑢((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
51	48, 49, 50	3bitr3i 210	. . . . . . 7 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
52	51	2exbii 1606	. . . . . 6 ⊢ (∃𝑧∃𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ∃𝑧∃𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
53		19.42vv 1911	. . . . . 6 ⊢ (∃𝑧∃𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
54		19.42vv 1911	. . . . . 6 ⊢ (∃𝑧∃𝑤((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
55	52, 53, 54	3bitr3i 210	. . . . 5 ⊢ (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
56	18, 55	bitri 184	. . . 4 ⊢ ((((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))) ↔ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))))
57	56	opabbii 4067	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((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)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
58		df-enq 7337	. . 3 ⊢ ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
59	57, 58	eqtr4i 2201	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ∧ (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)))} = ~Q
60	3, 4, 59	3eqtrri 2203	1 ⊢ ~Q = ( ~Q0 ∩ ((N × N) × (N × N)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ∩ cin 3128   ⊆ wss 3129  ⟨cop 3594  {copab 4060  ωcom 4586   × cxp 4621  (class class class)co 5869   ·_o comu 6409  Ncnpi 7262   ·_N cmi 7264   ~_Q ceq 7269   ~_Q0 ceq0 7276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-res 4635  df-iota 5174  df-fv 5220  df-ov 5872  df-ni 7294  df-mi 7296  df-enq 7337  df-enq0 7414

This theorem is referenced by:  nqnq0pi  7428