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Theorem nqpru 7328
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 ((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <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.
StepHypRef Expression
1 prop 7251 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7265 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
31, 2sylan 281 . . . . 5 ((𝐵P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
4 elprnqu 7258 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
51, 4sylan 281 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
65ad2ant2r 500 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥Q)
7 simprl 505 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd𝐵))
8 vex 2663 . . . . . . . . . . . 12 𝑥 ∈ V
9 breq1 3902 . . . . . . . . . . . 12 (𝑙 = 𝑥 → (𝑙 <Q 𝐴𝑥 <Q 𝐴))
108, 9elab 2802 . . . . . . . . . . 11 (𝑥 ∈ {𝑙𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
1110biimpri 132 . . . . . . . . . 10 (𝑥 <Q 𝐴𝑥 ∈ {𝑙𝑙 <Q 𝐴})
12 ltnqex 7325 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
13 gtnqex 7326 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1412, 13op1st 6012 . . . . . . . . . . 11 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
1514eleq2i 2184 . . . . . . . . . 10 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑙𝑙 <Q 𝐴})
1611, 15sylibr 133 . . . . . . . . 9 (𝑥 <Q 𝐴𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1716ad2antll 482 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
18 19.8a 1554 . . . . . . . 8 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
196, 7, 17, 18syl12anc 1199 . . . . . . 7 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
20 df-rex 2399 . . . . . . 7 (∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)) ↔ ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2119, 20sylibr 133 . . . . . 6 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))
22 elprnqu 7258 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
231, 22sylan 281 . . . . . . . 8 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
24 nqprlu 7323 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
25 ltdfpr 7282 . . . . . . . . 9 ((𝐵P ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2624, 25sylan2 284 . . . . . . . 8 ((𝐵P𝐴Q) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2723, 26syldan 280 . . . . . . 7 ((𝐵P𝐴 ∈ (2nd𝐵)) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2827adantr 274 . . . . . 6 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2921, 28mpbird 166 . . . . 5 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
303, 29rexlimddv 2531 . . . 4 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
3130ex 114 . . 3 (𝐵P → (𝐴 ∈ (2nd𝐵) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
3231adantl 275 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
3326ancoms 266 . . . . 5 ((𝐴Q𝐵P) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
3433biimpa 294 . . . 4 (((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))
3515, 10bitri 183 . . . . . . . 8 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐴)
3635biimpi 119 . . . . . . 7 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝑥 <Q 𝐴)
3736ad2antll 482 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))) → 𝑥 <Q 𝐴)
3837adantl 275 . . . . 5 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝑥 <Q 𝐴)
39 simpllr 508 . . . . . 6 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝐵P)
40 simprrl 513 . . . . . 6 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝑥 ∈ (2nd𝐵))
41 prcunqu 7261 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
421, 41sylan 281 . . . . . 6 ((𝐵P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
4339, 40, 42syl2anc 408 . . . . 5 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
4438, 43mpd 13 . . . 4 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝐴 ∈ (2nd𝐵))
4534, 44rexlimddv 2531 . . 3 (((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 ∈ (2nd𝐵))
4645ex 114 . 2 ((𝐴Q𝐵P) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ → 𝐴 ∈ (2nd𝐵)))
4732, 46impbid 128 1 ((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1453  wcel 1465  {cab 2103  wrex 2394  cop 3500   class class class wbr 3899  cfv 5093  1st c1st 6004  2nd c2nd 6005  Qcnq 7056   <Q cltq 7061  Pcnp 7067  <P cltp 7071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-iltp 7246
This theorem is referenced by:  prplnqu  7396  caucvgprprlemmu  7471  caucvgprprlemopu  7475  caucvgprprlemexbt  7482  caucvgprprlem2  7486  suplocexprlemloc  7497
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