Theorem nqnq0 7508

Description: A positive fraction is a nonnegative 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 7415	. . . . 5 ⊢ Q = ((N × N) / ~Q )
2	1	eleq2i 2263	. . . 4 ⊢ (𝑦 ∈ Q ↔ 𝑦 ∈ ((N × N) / ~Q ))
3		vex 2766	. . . . 5 ⊢ 𝑦 ∈ V
4	3	elqs 6645	. . . 4 ⊢ (𝑦 ∈ ((N × N) / ~Q ) ↔ ∃𝑥 ∈ (N × N)𝑦 = [𝑥] ~Q )
5		df-rex 2481	. . . 4 ⊢ (∃𝑥 ∈ (N × N)𝑦 = [𝑥] ~Q ↔ ∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ))
6	2, 4, 5	3bitri 206	. . 3 ⊢ (𝑦 ∈ Q ↔ ∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ))
7		elxpi 4679	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → ∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)))
8		nqnq0pi 7505	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q )
9	8	adantl 277	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q )
10		eceq1 6627	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q0 )
11		eceq1 6627	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q = [⟨𝑢, 𝑣⟩] ~Q )
12	10, 11	eqeq12d 2211	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → ([𝑥] ~Q0 = [𝑥] ~Q ↔ [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q ))
13	12	adantr 276	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → ([𝑥] ~Q0 = [𝑥] ~Q ↔ [⟨𝑢, 𝑣⟩] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q ))
14	9, 13	mpbird 167	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q0 = [𝑥] ~Q )
15		pinn 7376	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
16		opelxpi 4695	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ω ∧ 𝑣 ∈ N) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
17	15, 16	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
18	17	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → ⟨𝑢, 𝑣⟩ ∈ (ω × N))
19		eleq1 2259	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (𝑥 ∈ (ω × N) ↔ ⟨𝑢, 𝑣⟩ ∈ (ω × N)))
20	19	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → (𝑥 ∈ (ω × N) ↔ ⟨𝑢, 𝑣⟩ ∈ (ω × N)))
21	18, 20	mpbird 167	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → 𝑥 ∈ (ω × N))
22		enq0ex 7506	. . . . . . . . . . . 12 ⊢ ~Q0 ∈ V
23	22	ecelqsi 6648	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ω × N) → [𝑥] ~Q0 ∈ ((ω × N) / ~Q0 ))
24		df-nq0 7492	. . . . . . . . . . 11 ⊢ Q0 = ((ω × N) / ~Q0 )
25	23, 24	eleqtrrdi 2290	. . . . . . . . . 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 2274	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q ∈ Q0)
28	27	exlimivv 1911	. . . . . . 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 276	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → [𝑥] ~Q ∈ Q0)
31		eleq1 2259	. . . . . 6 ⊢ (𝑦 = [𝑥] ~Q → (𝑦 ∈ Q0 ↔ [𝑥] ~Q ∈ Q0))
32	31	adantl 277	. . . . 5 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → (𝑦 ∈ Q0 ↔ [𝑥] ~Q ∈ Q0))
33	30, 32	mpbird 167	. . . 4 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → 𝑦 ∈ Q0)
34	33	exlimiv 1612	. . 3 ⊢ (∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → 𝑦 ∈ Q0)
35	6, 34	sylbi 121	. 2 ⊢ (𝑦 ∈ Q → 𝑦 ∈ Q0)
36	35	ssriv 3187	1 ⊢ Q ⊆ Q0

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157  ⟨cop 3625  ωcom 4626   × cxp 4661  [cec 6590   / cqs 6591  Ncnpi 7339   ~_Q ceq 7346  Qcnq 7347   ~_Q0 ceq0 7353  Q₀cnq0 7354

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-mi 7373  df-enq 7414  df-nqqs 7415  df-enq0 7491  df-nq0 7492

This theorem is referenced by:  prarloclem5  7567  prarloclemcalc  7569