Theorem enq0ref 7234

Description: The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7236. (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 4550	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤(𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)))
2		elxpi 4550	. . . . . 6 ⊢ (𝑓 ∈ (ω × N) → ∃𝑣∃𝑢(𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)))
3		ee4anv 1904	. . . . . 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 413	. . . . 5 ⊢ (𝑓 ∈ (ω × N) → ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ ω ∧ 𝑤 ∈ N)) ∧ (𝑓 = ⟨𝑣, 𝑢⟩ ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N))))
5		eqtr2 2156	. . . . . . . . . . . 12 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩)
6		vex 2684	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
7		vex 2684	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
8	6, 7	opth 4154	. . . . . . . . . . . 12 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑣, 𝑢⟩ ↔ (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
9	5, 8	sylib 121	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → (𝑧 = 𝑣 ∧ 𝑤 = 𝑢))
10		oveq1 5774	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑣 → (𝑧 ·o 𝑢) = (𝑣 ·o 𝑢))
11		oveq2 5775	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑤 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
12	11	equcoms 1684	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑢 → (𝑣 ·o 𝑢) = (𝑣 ·o 𝑤))
13	10, 12	sylan9eq 2190	. . . . . . . . . . 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 321	. . . . . . . . 9 ⊢ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑣 ·o 𝑤)))
16	15	ad2ant2r 500	. . . . . . . 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 7110	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
18		nnmcom 6378	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ ω) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
19	17, 18	sylan2 284	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ N) → (𝑣 ·o 𝑤) = (𝑤 ·o 𝑣))
20	19	eqeq2d 2149	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ∈ N) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
21	20	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ ω) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
22	21	ad2ant2lr 501	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ((𝑧 ·o 𝑢) = (𝑣 ·o 𝑤) ↔ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))
23	22	ad2ant2l 499	. . . . . . . . 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 459	. . . . . . . 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 146	. . . . . . 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 1580	. . . . . 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 1580	. . . . 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 321	. . 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 2684	. . . . 5 ⊢ 𝑓 ∈ V
31		eleq1 2200	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
32	31	anbi1d 460	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
33		eqeq1 2144	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
34	33	anbi1d 460	. . . . . . . 8 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
35	34	anbi1d 460	. . . . . . 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 1842	. . . . . 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 464	. . . . 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 2200	. . . . . . 7 ⊢ (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
39	38	anbi2d 459	. . . . . 6 ⊢ (𝑦 = 𝑓 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
40		eqeq1 2144	. . . . . . . . 9 ⊢ (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
41	40	anbi2d 459	. . . . . . . 8 ⊢ (𝑦 = 𝑓 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
42	41	anbi1d 460	. . . . . . 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 1842	. . . . . 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 464	. . . . 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 7225	. . . . 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 4189	. . . 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 393	. . . . 5 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ↔ 𝑓 ∈ (ω × N))
48	47	anbi1i 453	. . . 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 183	. . 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 133	. 2 ⊢ (𝑓 ∈ (ω × N) → 𝑓 ~Q0 𝑓)
51	49	simplbi 272	. 2 ⊢ (𝑓 ~Q0 𝑓 → 𝑓 ∈ (ω × N))
52	50, 51	impbii 125	1 ⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ⟨cop 3525   class class class wbr 3924  ωcom 4499   × cxp 4532  (class class class)co 5767   ·_o comu 6304  Ncnpi 7073   ~_Q0 ceq0 7087

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-ni 7105  df-enq0 7225

This theorem is referenced by:  enq0er  7236