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Theorem nqpru 7090
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 7013 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7027 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
31, 2sylan 277 . . . . 5 ((𝐵P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
4 elprnqu 7020 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
51, 4sylan 277 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
65ad2ant2r 493 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥Q)
7 simprl 498 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd𝐵))
8 vex 2622 . . . . . . . . . . . 12 𝑥 ∈ V
9 breq1 3840 . . . . . . . . . . . 12 (𝑙 = 𝑥 → (𝑙 <Q 𝐴𝑥 <Q 𝐴))
108, 9elab 2758 . . . . . . . . . . 11 (𝑥 ∈ {𝑙𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
1110biimpri 131 . . . . . . . . . 10 (𝑥 <Q 𝐴𝑥 ∈ {𝑙𝑙 <Q 𝐴})
12 ltnqex 7087 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
13 gtnqex 7088 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1412, 13op1st 5899 . . . . . . . . . . 11 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
1514eleq2i 2154 . . . . . . . . . 10 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑙𝑙 <Q 𝐴})
1611, 15sylibr 132 . . . . . . . . 9 (𝑥 <Q 𝐴𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1716ad2antll 475 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
18 19.8a 1527 . . . . . . . 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 1172 . . . . . . 7 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
20 df-rex 2365 . . . . . . 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 132 . . . . . 6 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))
22 elprnqu 7020 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
231, 22sylan 277 . . . . . . . 8 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
24 nqprlu 7085 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
25 ltdfpr 7044 . . . . . . . . 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 280 . . . . . . . 8 ((𝐵P𝐴Q) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2723, 26syldan 276 . . . . . . 7 ((𝐵P𝐴 ∈ (2nd𝐵)) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
2827adantr 270 . . . . . 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 165 . . . . 5 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
303, 29rexlimddv 2493 . . . 4 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)
3130ex 113 . . 3 (𝐵P → (𝐴 ∈ (2nd𝐵) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
3231adantl 271 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) → 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
3326ancoms 264 . . . . 5 ((𝐴Q𝐵P) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
3433biimpa 290 . . . 4 (((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → ∃𝑥Q (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))
3515, 10bitri 182 . . . . . . . 8 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐴)
3635biimpi 118 . . . . . . 7 (𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝑥 <Q 𝐴)
3736ad2antll 475 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))) → 𝑥 <Q 𝐴)
3837adantl 271 . . . . 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 501 . . . . . 6 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝐵P)
40 simprrl 506 . . . . . 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 7023 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
421, 41sylan 277 . . . . . 6 ((𝐵P𝑥 ∈ (2nd𝐵)) → (𝑥 <Q 𝐴𝐴 ∈ (2nd𝐵)))
4339, 40, 42syl2anc 403 . . . . 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 2493 . . 3 (((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 ∈ (2nd𝐵))
4645ex 113 . 2 ((𝐴Q𝐵P) → (𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ → 𝐴 ∈ (2nd𝐵)))
4732, 46impbid 127 1 ((𝐴Q𝐵P) → (𝐴 ∈ (2nd𝐵) ↔ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1426  wcel 1438  {cab 2074  wrex 2360  cop 3444   class class class wbr 3837  cfv 5002  1st c1st 5891  2nd c2nd 5892  Qcnq 6818   <Q cltq 6823  Pcnp 6829  <P cltp 6833
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004  df-iltp 7008
This theorem is referenced by:  prplnqu  7158  caucvgprprlemmu  7233  caucvgprprlemopu  7237  caucvgprprlemexbt  7244  caucvgprprlem2  7248
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