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Theorem nqprl 6803
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 6727 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaxl 6740 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
31, 2sylan 277 . . . . 5 ((𝐵P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
4 elprnql 6733 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → 𝑥Q)
51, 4sylan 277 . . . . . . . . 9 ((𝐵P𝑥 ∈ (1st𝐵)) → 𝑥Q)
65ad2ant2r 493 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥Q)
7 vex 2605 . . . . . . . . . . . 12 𝑥 ∈ V
8 breq2 3797 . . . . . . . . . . . 12 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
97, 8elab 2739 . . . . . . . . . . 11 (𝑥 ∈ {𝑢𝐴 <Q 𝑢} ↔ 𝐴 <Q 𝑥)
109biimpri 131 . . . . . . . . . 10 (𝐴 <Q 𝑥𝑥 ∈ {𝑢𝐴 <Q 𝑢})
11 ltnqex 6801 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
12 gtnqex 6802 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1311, 12op2nd 5805 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1413eleq2i 2146 . . . . . . . . . 10 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑢𝐴 <Q 𝑢})
1510, 14sylibr 132 . . . . . . . . 9 (𝐴 <Q 𝑥𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1615ad2antll 475 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
17 simprl 498 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (1st𝐵))
18 19.8a 1523 . . . . . . . 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 1168 . . . . . . 7 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
20 df-rex 2355 . . . . . . 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 132 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
22 elprnql 6733 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → 𝐴Q)
231, 22sylan 277 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐴Q)
24 simpl 107 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐵P)
25 nqprlu 6799 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6758 . . . . . . . . 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 277 . . . . . . . 8 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2823, 24, 27syl2anc 403 . . . . . . 7 ((𝐵P𝐴 ∈ (1st𝐵)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2928adantr 270 . . . . . 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 165 . . . . 5 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
313, 30rexlimddv 2482 . . . 4 ((𝐵P𝐴 ∈ (1st𝐵)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
3231ex 113 . . 3 (𝐵P → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3332adantl 271 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3427biimpa 290 . . . 4 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
3514, 9bitri 182 . . . . . . . 8 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
3635biimpi 118 . . . . . . 7 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑥)
3736ad2antrl 474 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → 𝐴 <Q 𝑥)
3837adantl 271 . . . . 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 501 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐵P)
40 simprrr 507 . . . . . 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 6736 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
421, 41sylan 277 . . . . . 6 ((𝐵P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4339, 40, 42syl2anc 403 . . . . 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 2482 . . 3 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → 𝐴 ∈ (1st𝐵))
4645ex 113 . 2 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵𝐴 ∈ (1st𝐵)))
4733, 46impbid 127 1 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1422  wcel 1434  {cab 2068  wrex 2350  cop 3409   class class class wbr 3793  cfv 4932  1st c1st 5796  2nd c2nd 5797  Qcnq 6532   <Q cltq 6537  Pcnp 6543  <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-inp 6718  df-iltp 6722
This theorem is referenced by:  caucvgprlemcanl  6896  cauappcvgprlem1  6911  archrecpr  6916  caucvgprlem1  6931  caucvgprprlemml  6946  caucvgprprlemopl  6949
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