Theorem enq0ref 7493

Description: The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7495. (Contributed by Jim Kingdon, 14-Nov-2019.)

Assertion

Ref	Expression
enq0ref	⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Proof of Theorem enq0ref

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxpi 4675	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)))
2		elxpi 4675	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑣∃𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)))
3		ee4anv 1950	. . . . . 6 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) ↔ (∃𝑧∃𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ ∃𝑣∃𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))))
4	1, 2, 3	sylanbrc 417	. . . . 5 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))))
5		eqtr2 2212	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩)
6		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
7		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
8	6, 7	opth 4266	. . . . . . . . . . . 12 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
9	5, 8	sylib 122	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
10		oveq1 5925	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑢))
11		oveq2 5926	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
12	11	equcoms 1719	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
13	10, 12	sylan9eq 2246	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑣 ∧ 𝑤 = 𝑢) → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤))
14	9, 13	syl 14	. . . . . . . . . 10 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤))
15	14	ancli 323	. . . . . . . . 9 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)))
16	15	ad2ant2r 509	. . . . . . . 8 ⊢ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)))
17		pinn 7369	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
18		nnmcom 6542	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ ω) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
19	17, 18	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ N) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
20	19	eqeq2d 2205	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ N) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
21	20	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ ω) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
22	21	ad2ant2lr 510	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
23	22	ad2ant2l 508	. . . . . . . . 9 ⊢ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
24	23	anbi2d 464	. . . . . . . 8 ⊢ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
25	16, 24	mpbid 147	. . . . . . 7 ⊢ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
26	25	2eximi 1612	. . . . . 6 ⊢ (∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
27	26	2eximi 1612	. . . . 5 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))) → ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
28	4, 27	syl 14	. . . 4 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
29	28	ancli 323	. . 3 ⊢ (𝑓 ∈ (ω × N) → (𝑓 ∈ (ω × N) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
30		vex 2763	. . . . 5 ⊢ 𝑓 ∈ V
31		eleq1 2256	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
32	31	anbi1d 465	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
33		eqeq1 2200	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
34	33	anbi1d 465	. . . . . . . 8 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
35	34	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
36	35	4exbidv 1881	. . . . . 6 ⊢ (𝑥 = 𝑓 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
37	32, 36	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝑓 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
38		eleq1 2256	. . . . . . 7 ⊢ (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
39	38	anbi2d 464	. . . . . 6 ⊢ (𝑦 = 𝑓 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
40		eqeq1 2200	. . . . . . . . 9 ⊢ (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
41	40	anbi2d 464	. . . . . . . 8 ⊢ (𝑦 = 𝑓 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
42	41	anbi1d 465	. . . . . . 7 ⊢ (𝑦 = 𝑓 → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
43	42	4exbidv 1881	. . . . . 6 ⊢ (𝑦 = 𝑓 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
44	39, 43	anbi12d 473	. . . . 5 ⊢ (𝑦 = 𝑓 → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
45		df-enq0 7484	. . . . 5 ⊢ ~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)))}
46	30, 30, 37, 44, 45	brab 4303	. . . 4 ⊢ (𝑓 ~Q0 𝑓 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
47		anidm 396	. . . . 5 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ↔ 𝑓 ∈ (ω × N))
48	47	anbi1i 458	. . . 4 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ (𝑓 ∈ (ω × N) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
49	46, 48	bitri 184	. . 3 ⊢ (𝑓 ~Q0 𝑓 ↔ (𝑓 ∈ (ω × N) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
50	29, 49	sylibr 134	. 2 ⊢ (𝑓 ∈ (ω × N) → 𝑓 ~Q0 𝑓)
51	49	simplbi 274	. 2 ⊢ (𝑓 ~Q0 𝑓 → 𝑓 ∈ (ω × N))
52	50, 51	impbii 126	1 ⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  ωcom 4622   × cxp 4657  (class class class)co 5918   ·_o comu 6467  Ncnpi 7332   ~_Q0 ceq0 7346

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-ni 7364  df-enq0 7484

This theorem is referenced by:  enq0er  7495