Theorem nqnq0 6539

Description: A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)

Assertion

Ref	Expression
nqnq0	⊢ Q ⊆ Q0

Proof of Theorem nqnq0

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

Step	Hyp	Ref	Expression
1		df-nqqs 6446	. . . . 5 ⊢ Q = ((N × N) / ~Q )
2	1	eleq2i 2104	. . . 4 ⊢ (𝑦 ∈ Q ↔ 𝑦 ∈ ((N × N) / ~Q ))
3		vex 2560	. . . . 5 ⊢ 𝑦 ∈ V
4	3	elqs 6157	. . . 4 ⊢ (𝑦 ∈ ((N × N) / ~Q ) ↔ ∃𝑥 ∈ (N × N)𝑦 = [𝑥] ~Q )
5		df-rex 2312	. . . 4 ⊢ (∃𝑥 ∈ (N × N)𝑦 = [𝑥] ~Q ↔ ∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ))
6	2, 4, 5	3bitri 195	. . 3 ⊢ (𝑦 ∈ Q ↔ ∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ))
7		elxpi 4361	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → ∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)))
8		nqnq0pi 6536	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q )
9	8	adantl 262	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q )
10		eceq1 6141	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q0 )
11		eceq1 6141	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q = [⟨𝑢, 𝑣⟩] ~Q )
12	10, 11	eqeq12d 2054	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → ([𝑥] ~Q0 = [𝑥] ~Q ↔ [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q ))
13	12	adantr 261	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → ([𝑥] ~Q0 = [𝑥] ~Q ↔ [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q ))
14	9, 13	mpbird 156	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q0 = [𝑥] ~Q )
15		pinn 6407	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
16		opelxpi 4376	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ω ∧ 𝑣 ∈ N) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
17	15, 16	sylan 267	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
18	17	adantl 262	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
19		eleq1 2100	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥 ∈ (ω × N) ↔ ⟨𝑢, 𝑣⟩ ∈ (ω × N)))
20	19	adantr 261	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → (𝑥 ∈ (ω × N) ↔ ⟨𝑢, 𝑣⟩ ∈ (ω × N)))
21	18, 20	mpbird 156	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → 𝑥 ∈ (ω × N))
22		enq0ex 6537	. . . . . . . . . . . 12 ⊢ ~Q0 ∈ V
23	22	ecelqsi 6160	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ω × N) → [𝑥] ~Q0 ∈ ((ω × N) / ~Q0 ))
24		df-nq0 6523	. . . . . . . . . . 11 ⊢ Q0 = ((ω × N) / ~Q0 )
25	23, 24	syl6eleqr 2131	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ω × N) → [𝑥] ~Q0 ∈ Q0)
26	21, 25	syl 14	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q0 ∈ Q0)
27	14, 26	eqeltrrd 2115	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q ∈ Q0)
28	27	exlimivv 1776	. . . . . . 7 ⊢ (∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q ∈ Q0)
29	7, 28	syl 14	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → [𝑥] ~Q ∈ Q0)
30	29	adantr 261	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → [𝑥] ~Q ∈ Q0)
31		eleq1 2100	. . . . . 6 ⊢ (𝑦 = [𝑥] ~Q → (𝑦 ∈ Q0 ↔ [𝑥] ~Q ∈ Q0))
32	31	adantl 262	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → (𝑦 ∈ Q0 ↔ [𝑥] ~Q ∈ Q0))
33	30, 32	mpbird 156	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → 𝑦 ∈ Q0)
34	33	exlimiv 1489	. . 3 ⊢ (∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → 𝑦 ∈ Q0)
35	6, 34	sylbi 114	. 2 ⊢ (𝑦 ∈ Q → 𝑦 ∈ Q0)
36	35	ssriv 2949	1 ⊢ Q ⊆ Q0

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378  ωcom 4313   × cxp 4343  [cec 6104   / cqs 6105  Ncnpi 6370   ~_Q ceq 6377  Qcnq 6378   ~_Q0 ceq0 6384  Q₀cnq0 6385

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-enq 6445  df-nqqs 6446  df-enq0 6522  df-nq0 6523

This theorem is referenced by:  prarloclem5  6598  prarloclemcalc  6600