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Theorem nqprl 6706
Description: Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
Assertion
Ref Expression
nqprl ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
Distinct variable group:   𝐴,𝑙,𝑢
Allowed substitution hints:   𝐵(𝑢,𝑙)

Proof of Theorem nqprl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prop 6630 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaxl 6643 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
31, 2sylan 271 . . . . 5 ((𝐵P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
4 elprnql 6636 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → 𝑥Q)
51, 4sylan 271 . . . . . . . . 9 ((𝐵P𝑥 ∈ (1st𝐵)) → 𝑥Q)
65ad2ant2r 486 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥Q)
7 vex 2577 . . . . . . . . . . . 12 𝑥 ∈ V
8 breq2 3795 . . . . . . . . . . . 12 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
97, 8elab 2709 . . . . . . . . . . 11 (𝑥 ∈ {𝑢𝐴 <Q 𝑢} ↔ 𝐴 <Q 𝑥)
109biimpri 128 . . . . . . . . . 10 (𝐴 <Q 𝑥𝑥 ∈ {𝑢𝐴 <Q 𝑢})
11 ltnqex 6704 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
12 gtnqex 6705 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1311, 12op2nd 5801 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1413eleq2i 2120 . . . . . . . . . 10 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑢𝐴 <Q 𝑢})
1510, 14sylibr 141 . . . . . . . . 9 (𝐴 <Q 𝑥𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1615ad2antll 468 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
17 simprl 491 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (1st𝐵))
18 19.8a 1498 . . . . . . . 8 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
196, 16, 17, 18syl12anc 1144 . . . . . . 7 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
20 df-rex 2329 . . . . . . 7 (∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)) ↔ ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2119, 20sylibr 141 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
22 elprnql 6636 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → 𝐴Q)
231, 22sylan 271 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐴Q)
24 simpl 106 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐵P)
25 nqprlu 6702 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6661 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2725, 26sylan 271 . . . . . . . 8 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2823, 24, 27syl2anc 397 . . . . . . 7 ((𝐵P𝐴 ∈ (1st𝐵)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2928adantr 265 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
3021, 29mpbird 160 . . . . 5 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
313, 30rexlimddv 2454 . . . 4 ((𝐵P𝐴 ∈ (1st𝐵)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
3231ex 112 . . 3 (𝐵P → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3332adantl 266 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3427biimpa 284 . . . 4 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
3514, 9bitri 177 . . . . . . . 8 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
3635biimpi 117 . . . . . . 7 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑥)
3736ad2antrl 467 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → 𝐴 <Q 𝑥)
3837adantl 266 . . . . 5 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐴 <Q 𝑥)
39 simpllr 494 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐵P)
40 simprrr 500 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝑥 ∈ (1st𝐵))
41 prcdnql 6639 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
421, 41sylan 271 . . . . . 6 ((𝐵P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4339, 40, 42syl2anc 397 . . . . 5 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4438, 43mpd 13 . . . 4 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐴 ∈ (1st𝐵))
4534, 44rexlimddv 2454 . . 3 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → 𝐴 ∈ (1st𝐵))
4645ex 112 . 2 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵𝐴 ∈ (1st𝐵)))
4733, 46impbid 124 1 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wex 1397  wcel 1409  {cab 2042  wrex 2324  cop 3405   class class class wbr 3791  cfv 4929  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   <Q cltq 6440  Pcnp 6446  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-iltp 6625
This theorem is referenced by:  caucvgprlemcanl  6799  cauappcvgprlem1  6814  archrecpr  6819  caucvgprlem1  6834  caucvgprprlemml  6849  caucvgprprlemopl  6852
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