Theorem nqpru 7727

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 7650	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7664	. . . . . 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 7657	. . . . . . . . . 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 2802	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9		breq1 4085	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐴 ↔ 𝑥 <Q 𝐴))
10	8, 9	elab 2947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
11	10	biimpri 133	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
12		ltnqex 7724	. . . . . . . . . . . 12 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
13		gtnqex 7725	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
14	12, 13	op1st 6282	. . . . . . . . . . 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 7657	. . . . . . . . 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 7722	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
25		ltdfpr 7681	. . . . . . . . 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 7660	. . . . . . 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 3669   class class class wbr 4082  ‘cfv 5314  1^st c1st 6274  2^nd c2nd 6275  Qcnq 7455   <_Q cltq 7460  Pcnp 7466  <_P cltp 7470

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-inp 7641  df-iltp 7645

This theorem is referenced by:  prplnqu  7795  caucvgprprlemmu  7870  caucvgprprlemopu  7874  caucvgprprlemexbt  7881  caucvgprprlem2  7885  suplocexprlemloc  7896