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Theorem nqprl 7541
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 7465 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaxl 7478 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
31, 2sylan 283 . . . . 5 ((𝐵P𝐴 ∈ (1st𝐵)) → ∃𝑥 ∈ (1st𝐵)𝐴 <Q 𝑥)
4 elprnql 7471 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → 𝑥Q)
51, 4sylan 283 . . . . . . . . 9 ((𝐵P𝑥 ∈ (1st𝐵)) → 𝑥Q)
65ad2ant2r 509 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥Q)
7 vex 2740 . . . . . . . . . . . 12 𝑥 ∈ V
8 breq2 4004 . . . . . . . . . . . 12 (𝑢 = 𝑥 → (𝐴 <Q 𝑢𝐴 <Q 𝑥))
97, 8elab 2881 . . . . . . . . . . 11 (𝑥 ∈ {𝑢𝐴 <Q 𝑢} ↔ 𝐴 <Q 𝑥)
109biimpri 133 . . . . . . . . . 10 (𝐴 <Q 𝑥𝑥 ∈ {𝑢𝐴 <Q 𝑢})
11 ltnqex 7539 . . . . . . . . . . . 12 {𝑙𝑙 <Q 𝐴} ∈ V
12 gtnqex 7540 . . . . . . . . . . . 12 {𝑢𝐴 <Q 𝑢} ∈ V
1311, 12op2nd 6142 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
1413eleq2i 2244 . . . . . . . . . 10 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑥 ∈ {𝑢𝐴 <Q 𝑢})
1510, 14sylibr 134 . . . . . . . . 9 (𝐴 <Q 𝑥𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
1615ad2antll 491 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩))
17 simprl 529 . . . . . . . 8 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → 𝑥 ∈ (1st𝐵))
18 19.8a 1590 . . . . . . . 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 1236 . . . . . . 7 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥(𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
20 df-rex 2461 . . . . . . 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 134 . . . . . 6 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
22 elprnql 7471 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐴 ∈ (1st𝐵)) → 𝐴Q)
231, 22sylan 283 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐴Q)
24 simpl 109 . . . . . . . 8 ((𝐵P𝐴 ∈ (1st𝐵)) → 𝐵P)
25 nqprlu 7537 . . . . . . . . 9 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
26 ltdfpr 7496 . . . . . . . . 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 283 . . . . . . . 8 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2823, 24, 27syl2anc 411 . . . . . . 7 ((𝐵P𝐴 ∈ (1st𝐵)) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵 ↔ ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))))
2928adantr 276 . . . . . 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 167 . . . . 5 (((𝐵P𝐴 ∈ (1st𝐵)) ∧ (𝑥 ∈ (1st𝐵) ∧ 𝐴 <Q 𝑥)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
313, 30rexlimddv 2599 . . . 4 ((𝐵P𝐴 ∈ (1st𝐵)) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵)
3231ex 115 . . 3 (𝐵P → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3332adantl 277 . 2 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
3427biimpa 296 . . . 4 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → ∃𝑥Q (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))
3514, 9bitri 184 . . . . . . . 8 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
3635biimpi 120 . . . . . . 7 (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑥)
3736ad2antrl 490 . . . . . 6 ((𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵))) → 𝐴 <Q 𝑥)
3837adantl 277 . . . . 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 534 . . . . . 6 ((((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) ∧ (𝑥Q ∧ (𝑥 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st𝐵)))) → 𝐵P)
40 simprrr 540 . . . . . 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 7474 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
421, 41sylan 283 . . . . . 6 ((𝐵P𝑥 ∈ (1st𝐵)) → (𝐴 <Q 𝑥𝐴 ∈ (1st𝐵)))
4339, 40, 42syl2anc 411 . . . . 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 2599 . . 3 (((𝐴Q𝐵P) ∧ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵) → 𝐴 ∈ (1st𝐵))
4645ex 115 . 2 ((𝐴Q𝐵P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵𝐴 ∈ (1st𝐵)))
4733, 46impbid 129 1 ((𝐴Q𝐵P) → (𝐴 ∈ (1st𝐵) ↔ ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩<P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1492  wcel 2148  {cab 2163  wrex 2456  cop 3594   class class class wbr 4000  cfv 5212  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   <Q cltq 7275  Pcnp 7281  <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  caucvgprlemcanl  7634  cauappcvgprlem1  7649  archrecpr  7654  caucvgprlem1  7669  caucvgprprlemml  7684  caucvgprprlemopl  7687
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