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Theorem nqprl 6532
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 ((A Q B P) → (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 6457 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
2 prnmaxl 6470 . . . . . 6 ((⟨(1stB), (2ndB)⟩ P A (1stB)) → x (1stB)A <Q x)
31, 2sylan 267 . . . . 5 ((B P A (1stB)) → x (1stB)A <Q x)
4 elprnql 6463 . . . . . . . . . 10 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → x Q)
51, 4sylan 267 . . . . . . . . 9 ((B P x (1stB)) → x Q)
65ad2ant2r 478 . . . . . . . 8 (((B P A (1stB)) (x (1stB) A <Q x)) → x Q)
7 vex 2554 . . . . . . . . . . . 12 x V
8 breq2 3759 . . . . . . . . . . . 12 (u = x → (A <Q uA <Q x))
97, 8elab 2681 . . . . . . . . . . 11 (x {uA <Q u} ↔ A <Q x)
109biimpri 124 . . . . . . . . . 10 (A <Q xx {uA <Q u})
11 ltnqex 6530 . . . . . . . . . . . 12 {𝑙𝑙 <Q A} V
12 gtnqex 6531 . . . . . . . . . . . 12 {uA <Q u} V
1311, 12op2nd 5716 . . . . . . . . . . 11 (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) = {uA <Q u}
1413eleq2i 2101 . . . . . . . . . 10 (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) ↔ x {uA <Q u})
1510, 14sylibr 137 . . . . . . . . 9 (A <Q xx (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩))
1615ad2antll 460 . . . . . . . 8 (((B P A (1stB)) (x (1stB) A <Q x)) → x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩))
17 simprl 483 . . . . . . . 8 (((B P A (1stB)) (x (1stB) A <Q x)) → x (1stB))
18 19.8a 1479 . . . . . . . 8 ((x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))) → x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
196, 16, 17, 18syl12anc 1132 . . . . . . 7 (((B P A (1stB)) (x (1stB) A <Q x)) → x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
20 df-rex 2306 . . . . . . 7 (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)) ↔ x(x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2119, 20sylibr 137 . . . . . 6 (((B P A (1stB)) (x (1stB) A <Q x)) → x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))
22 elprnql 6463 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P A (1stB)) → A Q)
231, 22sylan 267 . . . . . . . 8 ((B P A (1stB)) → A Q)
24 simpl 102 . . . . . . . 8 ((B P A (1stB)) → B P)
25 nqprlu 6529 . . . . . . . . 9 (A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
26 ltdfpr 6488 . . . . . . . . 9 ((⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2725, 26sylan 267 . . . . . . . 8 ((A Q B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2823, 24, 27syl2anc 391 . . . . . . 7 ((B P A (1stB)) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
2928adantr 261 . . . . . 6 (((B P A (1stB)) (x (1stB) A <Q x)) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P Bx Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))))
3021, 29mpbird 156 . . . . 5 (((B P A (1stB)) (x (1stB) A <Q x)) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B)
313, 30rexlimddv 2431 . . . 4 ((B P A (1stB)) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B)
3231ex 108 . . 3 (B P → (A (1stB) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))
3332adantl 262 . 2 ((A Q B P) → (A (1stB) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))
3427biimpa 280 . . . 4 (((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) → x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))
3514, 9bitri 173 . . . . . . . 8 (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) ↔ A <Q x)
3635biimpi 113 . . . . . . 7 (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) → A <Q x)
3736ad2antrl 459 . . . . . 6 ((x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB))) → A <Q x)
3837adantl 262 . . . . 5 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → A <Q x)
39 simpllr 486 . . . . . 6 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → B P)
40 simprrr 492 . . . . . 6 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → x (1stB))
41 prcdnql 6466 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → (A <Q xA (1stB)))
421, 41sylan 267 . . . . . 6 ((B P x (1stB)) → (A <Q xA (1stB)))
4339, 40, 42syl2anc 391 . . . . 5 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → (A <Q xA (1stB)))
4438, 43mpd 13 . . . 4 ((((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) (x Q (x (2nd ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) x (1stB)))) → A (1stB))
4534, 44rexlimddv 2431 . . 3 (((A Q B P) ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B) → A (1stB))
4645ex 108 . 2 ((A Q B P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P BA (1stB)))
4733, 46impbid 120 1 ((A Q B P) → (A (1stB) ↔ ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩<P B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  {cab 2023  wrex 2301  cop 3370   class class class wbr 3755  cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   <Q cltq 6269  Pcnp 6275  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-iltp 6452
This theorem is referenced by:  cauappcvgprlemcan  6615  cauappcvgprlem1  6630
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