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Theorem nqprl 7057
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 6981 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaxl 6994 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
31, 2sylan 277 . . . . 5 ((𝐵P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
4 elprnql 6987 . . . . . . . . . 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 2618 . . . . . . . . . . . 12 𝑥 ∈ V
8 breq2 3826 . . . . . . . . . . . 12 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
97, 8elab 2751 . . . . . . . . . . 11 (𝑥 ∈ {𝑢𝐴 <Q 𝑢} ↔ 𝐴 <Q 𝑥)
109biimpri 131 . . . . . . . . . 10 (𝐴 <Q 𝑥𝑥 ∈ {𝑢𝐴 <Q 𝑢})
11 ltnqex 7055 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
12 gtnqex 7056 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1311, 12op2nd 5877 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1413eleq2i 2151 . . . . . . . . . 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 1525 . . . . . . . 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 1170 . . . . . . 7 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
20 df-rex 2361 . . . . . . 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 6987 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → 𝐴Q)
231, 22sylan 277 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐴Q)
24 simpl 107 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐵P)
25 nqprlu 7053 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
26 ltdfpr 7012 . . . . . . . . 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 2489 . . . 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 6990 . . . . . . 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 2489 . . 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 1424  wcel 1436  {cab 2071  wrex 2356  cop 3434   class class class wbr 3822  cfv 4983  1st c1st 5868  2nd c2nd 5869  Qcnq 6786   <Q cltq 6791  Pcnp 6797  <P cltp 6801
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-inp 6972  df-iltp 6976
This theorem is referenced by:  caucvgprlemcanl  7150  cauappcvgprlem1  7165  archrecpr  7170  caucvgprlem1  7185  caucvgprprlemml  7200  caucvgprprlemopl  7203
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