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Theorem nqpru 7514
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 7437 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7451 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
31, 2sylan 281 . . . . 5 ((𝐵P𝐴 ∈ (2nd𝐵)) → ∃𝑥 ∈ (2nd𝐵)𝑥 <Q 𝐴)
4 elprnqu 7444 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
51, 4sylan 281 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
65ad2ant2r 506 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥Q)
7 simprl 526 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (2nd𝐵))
8 vex 2733 . . . . . . . . . . . 12 𝑥 ∈ V
9 breq1 3992 . . . . . . . . . . . 12 (𝑙 = 𝑥 → (𝑙 <Q 𝐴𝑥 <Q 𝐴))
108, 9elab 2874 . . . . . . . . . . 11 (𝑥 ∈ {𝑙𝑙 <Q 𝐴} ↔ 𝑥 <Q 𝐴)
1110biimpri 132 . . . . . . . . . 10 (𝑥 <Q 𝐴𝑥 ∈ {𝑙𝑙 <Q 𝐴})
12 ltnqex 7511 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
13 gtnqex 7512 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1412, 13op1st 6125 . . . . . . . . . . 11 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
1514eleq2i 2237 . . . . . . . . . 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 488 . . . . . . . 8 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
18 19.8a 1583 . . . . . . . 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 1231 . . . . . . 7 (((𝐵P𝐴 ∈ (2nd𝐵)) ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 <Q 𝐴)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))))
20 df-rex 2454 . . . . . . 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 7444 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
231, 22sylan 281 . . . . . . . 8 ((𝐵P𝐴 ∈ (2nd𝐵)) → 𝐴Q)
24 nqprlu 7509 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
25 ltdfpr 7468 . . . . . . . . 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 2592 . . . 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 488 . . . . . 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 529 . . . . . 6 ((((𝐴Q𝐵P) ∧ 𝐵<P ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd𝐵) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)))) → 𝐵P)
40 simprrl 534 . . . . . 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 7447 . . . . . . 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 2592 . . 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 1485  wcel 2141  {cab 2156  wrex 2449  cop 3586   class class class wbr 3989  cfv 5198  1st c1st 6117  2nd c2nd 6118  Qcnq 7242   <Q cltq 7247  Pcnp 7253  <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432
This theorem is referenced by:  prplnqu  7582  caucvgprprlemmu  7657  caucvgprprlemopu  7661  caucvgprprlemexbt  7668  caucvgprprlem2  7672  suplocexprlemloc  7683
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