Theorem enq0ref 7588

Description: The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7590. (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 4712	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)))
2		elxpi 4712	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑣∃𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)))
3		ee4anv 1965	. . . . . 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 2228	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩)
6		vex 2782	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
7		vex 2782	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
8	6, 7	opth 4302	. . . . . . . . . . . 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 5981	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑢))
11		oveq2 5982	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
12	11	equcoms 1734	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
13	10, 12	sylan9eq 2262	. . . . . . . . . . 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 7464	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
18		nnmcom 6605	. . . . . . . . . . . . . 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 2221	. . . . . . . . . . . 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 1627	. . . . . 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 1627	. . . . 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 2782	. . . . 5 ⊢ 𝑓 ∈ V
31		eleq1 2272	. . . . . . 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 2216	. . . . . . . . 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 1896	. . . . . 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 2272	. . . . . . 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 2216	. . . . . . . . 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 1896	. . . . . 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 7579	. . . . 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 4340	. . . 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 1375  ∃wex 1518   ∈ wcel 2180  ⟨cop 3649   class class class wbr 4062  ωcom 4659   × cxp 4694  (class class class)co 5974   ·_o comu 6530  Ncnpi 7427   ~_Q0 ceq0 7441

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-oadd 6536  df-omul 6537  df-ni 7459  df-enq0 7579

This theorem is referenced by:  enq0er  7590