Theorem nqnq0 6296

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 6207	. . . . 5 ⊢ Q = ((N × N) / ~Q )
2	1	eleq2i 2086	. . . 4 ⊢ (y ∈ Q ↔ y ∈ ((N × N) / ~Q ))
3		vex 2538	. . . . 5 ⊢ y ∈ V
4	3	elqs 6068	. . . 4 ⊢ (y ∈ ((N × N) / ~Q ) ↔ ∃x ∈ (N × N)y = [x] ~Q )
5		df-rex 2290	. . . 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 4288	. . . . . . 7 ⊢ (x ∈ (N × N) → ∃u∃v(x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)))
8		nqnq0pi 6293	. . . . . . . . . . 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 6052	. . . . . . . . . . . 12 ⊢ (x = ⟨u, v⟩ → [x] ~Q0 = [⟨u, v⟩] ~Q0 )
11		eceq1 6052	. . . . . . . . . . . 12 ⊢ (x = ⟨u, v⟩ → [x] ~Q = [⟨u, v⟩] ~Q )
12	10, 11	eqeq12d 2036	. . . . . . . . . . 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 6169	. . . . . . . . . . . . 13 ⊢ (u ∈ N → u ∈ 𝜔)
16		opelxpi 4303	. . . . . . . . . . . . 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 2082	. . . . . . . . . . . 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 6294	. . . . . . . . . . . 12 ⊢ ~Q0 ∈ V
23	22	ecelqsi 6071	. . . . . . . . . . 11 ⊢ (x ∈ (𝜔 × N) → [x] ~Q0 ∈ ((𝜔 × N) / ~Q0 ))
24		df-nq0 6280	. . . . . . . . . . 11 ⊢ Q0 = ((𝜔 × N) / ~Q0 )
25	23, 24	syl6eleqr 2113	. . . . . . . . . 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 2097	. . . . . . . 8 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [x] ~Q ∈ Q0)
28	27	exlimivv 1758	. . . . . . 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 2082	. . . . . 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 1471	. . 3 ⊢ (∃x(x ∈ (N × N) ∧ y = [x] ~Q ) → y ∈ Q0)
35	6, 34	sylbi 114	. 2 ⊢ (y ∈ Q → y ∈ Q0)
36	35	ssriv 2926	1 ⊢ Q ⊆ Q0

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃wrex 2285   ⊆ wss 2894  ⟨cop 3353  𝜔com 4240   × cxp 4270  [cec 6015   / cqs 6016  Ncnpi 6130   ~_Q ceq 6137  Qcnq 6138   ~_Q0 ceq0 6144  Q₀cnq0 6145

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-mi 6166  df-enq 6206  df-nqqs 6207  df-enq0 6279  df-nq0 6280

This theorem is referenced by:  prarloclem5  6354  prarloclemcalc  6356