Theorem nqpru 7384

Description: Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <_P. (Contributed by Jim Kingdon, 29-Nov-2020.)

Assertion

Ref	Expression
nqpru	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))

Distinct variable group:   𝐴,𝑙,𝑢

Allowed substitution hints:   𝐵(𝑢,𝑙)

Proof of Theorem nqpru

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7307	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7321	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑥 <Q 𝐴)
3	1, 2	sylan 281	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑥 <Q 𝐴)
4		elprnqu 7314	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
5	1, 4	sylan 281	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
6	5	ad2ant2r 501	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ Q)
7		simprl 521	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd ‘𝐵))
8		vex 2692	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9		breq1 3940	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐴 ↔ 𝑥 <Q 𝐴))
10	8, 9	elab 2832	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
11	10	biimpri 132	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
12		ltnqex 7381	. . . . . . . . . . . 12 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
13		gtnqex 7382	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
14	12, 13	op1st 6052	. . . . . . . . . . 11 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐴}
15	14	eleq2i 2207	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
16	11, 15	sylibr 133	. . . . . . . . 9 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
17	16	ad2antll 483	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
18		19.8a 1570	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))) → ∃𝑥(𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
19	6, 7, 17, 18	syl12anc 1215	. . . . . . 7 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥(𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
20		df-rex 2423	. . . . . . 7 ⊢ (∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)) ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
21	19, 20	sylibr 133	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))
22		elprnqu 7314	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐴 ∈ Q)
23	1, 22	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐴 ∈ Q)
24		nqprlu 7379	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
25		ltdfpr 7338	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
26	24, 25	sylan2 284	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ Q) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
27	23, 26	syldan 280	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
28	27	adantr 274	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
29	21, 28	mpbird 166	. . . . 5 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
30	3, 29	rexlimddv 2557	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
31	30	ex 114	. . 3 ⊢ (𝐵 ∈ P → (𝐴 ∈ (2nd ‘𝐵) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
32	31	adantl 275	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
33	26	ancoms 266	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))))
34	33	biimpa 294	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))
35	15, 10	bitri 183	. . . . . . . 8 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐴)
36	35	biimpi 119	. . . . . . 7 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝑥 <Q 𝐴)
37	36	ad2antll 483	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))) → 𝑥 <Q 𝐴)
38	37	adantl 275	. . . . 5 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))) → 𝑥 <Q 𝐴)
39		simpllr 524	. . . . . 6 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))) → 𝐵 ∈ P)
40		simprrl 529	. . . . . 6 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))) → 𝑥 ∈ (2nd ‘𝐵))
41		prcunqu 7317	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → (𝑥 <Q 𝐴 → 𝐴 ∈ (2nd ‘𝐵)))
42	1, 41	sylan 281	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → (𝑥 <Q 𝐴 → 𝐴 ∈ (2nd ‘𝐵)))
43	39, 40, 42	syl2anc 409	. . . . 5 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))) → (𝑥 <Q 𝐴 → 𝐴 ∈ (2nd ‘𝐵)))
44	38, 43	mpd 13	. . . 4 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ (𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)))) → 𝐴 ∈ (2nd ‘𝐵))
45	34, 44	rexlimddv 2557	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝐴 ∈ (2nd ‘𝐵))
46	45	ex 114	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ → 𝐴 ∈ (2nd ‘𝐵)))
47	32, 46	impbid 128	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∃wrex 2418  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  1^st c1st 6044  2^nd c2nd 6045  Qcnq 7112   <_Q cltq 7117  Pcnp 7123  <_P cltp 7127

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298  df-iltp 7302

This theorem is referenced by:  prplnqu  7452  caucvgprprlemmu  7527  caucvgprprlemopu  7531  caucvgprprlemexbt  7538  caucvgprprlem2  7542  suplocexprlemloc  7553