ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nqpru GIF version

Theorem nqpru 7472
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 7395 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7409 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
31, 2sylan 281 . . . . 5 ((𝐵P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
4 elprnqu 7402 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
51, 4sylan 281 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
65ad2ant2r 501 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥Q)
7 simprl 521 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd𝐵))
8 vex 2715 . . . . . . . . . . . 12 𝑥 ∈ V
9 breq1 3968 . . . . . . . . . . . 12 (𝑙 = 𝑥 → (𝑙 <Q 𝐴𝑥 <Q 𝐴))
108, 9elab 2856 . . . . . . . . . . 11 (𝑥 ∈ {𝑙𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
1110biimpri 132 . . . . . . . . . 10 (𝑥 <Q 𝐴𝑥 ∈ {𝑙𝑙 <Q 𝐴})
12 ltnqex 7469 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
13 gtnqex 7470 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1412, 13op1st 6094 . . . . . . . . . . 11 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
1514eleq2i 2224 . . . . . . . . . 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 483 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
18 19.8a 1570 . . . . . . . 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 1218 . . . . . . 7 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
20 df-rex 2441 . . . . . . 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 7402 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
231, 22sylan 281 . . . . . . . 8 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
24 nqprlu 7467 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
25 ltdfpr 7426 . . . . . . . . 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 2579 . . . 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 483 . . . . . 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 524 . . . . . 6 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝐵P)
40 simprrl 529 . . . . . 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 7405 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
421, 41sylan 281 . . . . . 6 ((𝐵P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
4339, 40, 42syl2anc 409 . . . . 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 2579 . . 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 1472  wcel 2128  {cab 2143  wrex 2436  cop 3563   class class class wbr 3965  cfv 5170  1st c1st 6086  2nd c2nd 6087  Qcnq 7200   <Q cltq 7205  Pcnp 7211  <P cltp 7215
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-inp 7386  df-iltp 7390
This theorem is referenced by:  prplnqu  7540  caucvgprprlemmu  7615  caucvgprprlemopu  7619  caucvgprprlemexbt  7626  caucvgprprlem2  7630  suplocexprlemloc  7641
  Copyright terms: Public domain W3C validator