Theorem nqpru 7550

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 7473	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7487	. . . . . 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 7480	. . . . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9		breq1 4006	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐴 ↔ 𝑥 <Q 𝐴))
10	8, 9	elab 2881	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
11	10	biimpri 133	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝐴 → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q 𝐴})
12		ltnqex 7547	. . . . . . . . . . . 12 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
13		gtnqex 7548	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
14	12, 13	op1st 6146	. . . . . . . . . . 11 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐴}
15	14	eleq2i 2244	. . . . . . . . . 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 1590	. . . . . . . 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 1236	. . . . . . 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 2461	. . . . . . 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 7480	. . . . . . . . 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 7545	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
25		ltdfpr 7504	. . . . . . . . 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 2599	. . . 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 7483	. . . . . . 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 2599	. . 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 1492   ∈ wcel 2148  {cab 2163  ∃wrex 2456  ⟨cop 3595   class class class wbr 4003  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139  Qcnq 7278   <_Q cltq 7283  Pcnp 7289  <_P cltp 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-iltp 7468

This theorem is referenced by:  prplnqu  7618  caucvgprprlemmu  7693  caucvgprprlemopu  7697  caucvgprprlemexbt  7704  caucvgprprlem2  7708  suplocexprlemloc  7719