Theorem nqnq0 7802

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 7709	. . . . 5 ⊢ Q = ((N × N) / ~Q )
2	1	eleq2i 2305	. . . 4 ⊢ (𝑦 ∈ Q ↔ 𝑦 ∈ ((N × N) / ~Q ))
3		vex 2824	. . . . 5 ⊢ 𝑦 ∈ V
4	3	elqs 6854	. . . 4 ⊢ (𝑦 ∈ ((N × N) / ~Q ) ↔ ∃𝑥 ∈ (N × N)𝑦 = [𝑥] ~Q )
5		df-rex 2534	. . . 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 4788	. . . . . . 7 ⊢ (𝑥 ∈ (N × N) → ∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)))
8		nqnq0pi 7799	. . . . . . . . . . 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 6836	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q0 = [⟨𝑢, 𝑣⟩] ~Q0 )
11		eceq1 6836	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → [𝑥] ~Q = [⟨𝑢, 𝑣⟩] ~Q )
12	10, 11	eqeq12d 2253	. . . . . . . . . . 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 7670	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
16		opelxpi 4804	. . . . . . . . . . . . 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 2301	. . . . . . . . . . . 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 7800	. . . . . . . . . . . 12 ⊢ ~Q0 ∈ V
23	22	ecelqsi 6857	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ω × N) → [𝑥] ~Q0 ∈ ((ω × N) / ~Q0 ))
24		df-nq0 7786	. . . . . . . . . . 11 ⊢ Q0 = ((ω × N) / ~Q0 )
25	23, 24	eleqtrrdi 2332	. . . . . . . . . 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 2316	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ N ∧ 𝑣 ∈ N)) → [𝑥] ~Q ∈ Q0)
28	27	exlimivv 1952	. . . . . . 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 2301	. . . . . 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 1651	. . 3 ⊢ (∃𝑥(𝑥 ∈ (N × N) ∧ 𝑦 = [𝑥] ~Q ) → 𝑦 ∈ Q0)
35	6, 34	sylbi 121	. 2 ⊢ (𝑦 ∈ Q → 𝑦 ∈ Q0)
36	35	ssriv 3252	1 ⊢ Q ⊆ Q0

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3711  ωcom 4735   × cxp 4770  [cec 6799   / cqs 6800  Ncnpi 7633   ~_Q ceq 7640  Qcnq 7641   ~_Q0 ceq0 7647  Q₀cnq0 7648

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-mi 7667  df-enq 7708  df-nqqs 7709  df-enq0 7785  df-nq0 7786

This theorem is referenced by:  prarloclem5  7861  prarloclemcalc  7863