Theorem nqpru 7765

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 7688	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7702	. . . . . 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 7695	. . . . . . . . . 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 509	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ Q)
7		simprl 529	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐴 ∈ (2nd ‘𝐵)) ∧ (𝑥 ∈ (2nd ‘𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd ‘𝐵))
8		vex 2803	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9		breq1 4089	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐴 ↔ 𝑥 <Q 𝐴))
10	8, 9	elab 2948	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
11	10	biimpri 133	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
12		ltnqex 7762	. . . . . . . . . . . 12 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
13		gtnqex 7763	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
14	12, 13	op1st 6304	. . . . . . . . . . 11 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐴}
15	14	eleq2i 2296	. . . . . . . . . 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 491	. . . . . . . 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 1636	. . . . . . . 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 1269	. . . . . . 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 2514	. . . . . . 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 7695	. . . . . . . . 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 7760	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
25		ltdfpr 7719	. . . . . . . . 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 2653	. . . 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 491	. . . . . 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 534	. . . . . 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 539	. . . . . 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 7698	. . . . . . 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 411	. . . . 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 2653	. . 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}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∃wrex 2509  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Qcnq 7493   <_Q cltq 7498  Pcnp 7504  <_P cltp 7508

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-inp 7679  df-iltp 7683

This theorem is referenced by:  prplnqu  7833  caucvgprprlemmu  7908  caucvgprprlemopu  7912  caucvgprprlemexbt  7919  caucvgprprlem2  7923  suplocexprlemloc  7934