Theorem nqpru 7926

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 7849	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7863	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑥 <Q 𝐴)
3	1, 2	sylan 283	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → ∃𝑥 ∈ (2nd ‘𝐵)𝑥 <Q 𝐴)
4		elprnqu 7856	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
5	1, 4	sylan 283	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → 𝑥 ∈ Q)
6	5	ad2ant2r 513	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ Q)
7		simprl 535	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd ‘𝐵))
8		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9		breq1 4134	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐴 ↔ 𝑥 <Q 𝐴))
10	8, 9	elab 2970	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
11	10	biimpri 133	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
12		ltnqex 7923	. . . . . . . . . . . 12 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
13		gtnqex 7924	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
14	12, 13	op1st 6384	. . . . . . . . . . 11 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐴}
15	14	eleq2i 2305	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
16	11, 15	sylibr 134	. . . . . . . . 9 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
17	16	ad2antll 495	. . . . . . . 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 1643	. . . . . . . 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 1276	. . . . . . 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 2534	. . . . . . 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 134	. . . . . 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 7856	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐴 ∈ Q)
23	1, 22	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐴 ∈ Q)
24		nqprlu 7921	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
25		ltdfpr 7880	. . . . . . . . 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 286	. . . . . . . 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 282	. . . . . . 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 276	. . . . . 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 167	. . . . 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 2673	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
31	30	ex 115	. . 3 ⊢ (𝐵 ∈ P → (𝐴 ∈ (2nd ‘𝐵) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
32	31	adantl 277	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) → 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
33	26	ancoms 268	. . . . 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 296	. . . 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 184	. . . . . . . 8 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐴)
36	35	biimpi 120	. . . . . . 7 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝑥 <Q 𝐴)
37	36	ad2antll 495	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))) → 𝑥 <Q 𝐴)
38	37	adantl 277	. . . . 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 540	. . . . . 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 545	. . . . . 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 7859	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → (𝑥 <Q 𝐴 → 𝐴 ∈ (2nd ‘𝐵)))
42	1, 41	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐵)) → (𝑥 <Q 𝐴 → 𝐴 ∈ (2nd ‘𝐵)))
43	39, 40, 42	syl2anc 415	. . . . 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 2673	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ P) ∧ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝐴 ∈ (2nd ‘𝐵))
46	45	ex 115	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ → 𝐴 ∈ (2nd ‘𝐵)))
47	32, 46	impbid 129	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  ⟨cop 3712   class class class wbr 4131  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377  Qcnq 7654   <_Q cltq 7659  Pcnp 7665  <_P cltp 7669

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7678  df-pli 7679  df-mi 7680  df-lti 7681  df-plpq 7718  df-mpq 7719  df-enq 7721  df-nqqs 7722  df-plqqs 7723  df-mqqs 7724  df-1nqqs 7725  df-rq 7726  df-ltnqqs 7727  df-inp 7840  df-iltp 7844

This theorem is used by:  prplnqu  7994  caucvgprprlemmu  8069  caucvgprprlemopu  8073  caucvgprprlemexbt  8080  caucvgprprlem2  8084  suplocexprlemloc  8095