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Theorem ltexprlemlol 6576
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
Assertion
Ref Expression
ltexprlemlol ((A<P B 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) → 𝑞 (1st𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 482 . . . . . 6 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → 𝑞 Q)
2 simprrr 492 . . . . . . 7 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → (y (2ndA) (y +Q 𝑟) (1stB)))
32simpld 105 . . . . . 6 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → y (2ndA))
4 simprl 483 . . . . . . . 8 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → 𝑞 <Q 𝑟)
5 simpll 481 . . . . . . . . 9 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → A<P B)
6 ltrelpr 6488 . . . . . . . . . . . 12 <P ⊆ (P × P)
76brel 4335 . . . . . . . . . . 11 (A<P B → (A P B P))
87simpld 105 . . . . . . . . . 10 (A<P BA P)
9 prop 6458 . . . . . . . . . . 11 (A P → ⟨(1stA), (2ndA)⟩ P)
10 elprnqu 6465 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
119, 10sylan 267 . . . . . . . . . 10 ((A P y (2ndA)) → y Q)
128, 11sylan 267 . . . . . . . . 9 ((A<P B y (2ndA)) → y Q)
135, 3, 12syl2anc 391 . . . . . . . 8 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → y Q)
14 ltanqi 6386 . . . . . . . 8 ((𝑞 <Q 𝑟 y Q) → (y +Q 𝑞) <Q (y +Q 𝑟))
154, 13, 14syl2anc 391 . . . . . . 7 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → (y +Q 𝑞) <Q (y +Q 𝑟))
167simprd 107 . . . . . . . . 9 (A<P BB P)
175, 16syl 14 . . . . . . . 8 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → B P)
182simprd 107 . . . . . . . 8 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → (y +Q 𝑟) (1stB))
19 prop 6458 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
20 prcdnql 6467 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P (y +Q 𝑟) (1stB)) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑞) (1stB)))
2119, 20sylan 267 . . . . . . . 8 ((B P (y +Q 𝑟) (1stB)) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑞) (1stB)))
2217, 18, 21syl2anc 391 . . . . . . 7 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → ((y +Q 𝑞) <Q (y +Q 𝑟) → (y +Q 𝑞) (1stB)))
2315, 22mpd 13 . . . . . 6 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → (y +Q 𝑞) (1stB))
241, 3, 23jca32 293 . . . . 5 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))))
2524eximi 1488 . . . 4 (y((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) → y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))))
26 ltexprlem.1 . . . . . . . . . 10 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
2726ltexprlemell 6572 . . . . . . . . 9 (𝑟 (1st𝐶) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
28 19.42v 1783 . . . . . . . . 9 (y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
2927, 28bitr4i 176 . . . . . . . 8 (𝑟 (1st𝐶) ↔ y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))
3029anbi2i 430 . . . . . . 7 ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ (𝑞 <Q 𝑟 y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
31 19.42v 1783 . . . . . . 7 (y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
3230, 31bitr4i 176 . . . . . 6 ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
3332anbi2i 430 . . . . 5 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 𝑟 (1st𝐶))) ↔ ((A<P B 𝑞 Q) y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
34 19.42v 1783 . . . . 5 (y((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))) ↔ ((A<P B 𝑞 Q) y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
3533, 34bitr4i 176 . . . 4 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 𝑟 (1st𝐶))) ↔ y((A<P B 𝑞 Q) (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
3626ltexprlemell 6572 . . . . 5 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
37 19.42v 1783 . . . . 5 (y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
3836, 37bitr4i 176 . . . 4 (𝑞 (1st𝐶) ↔ y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))))
3925, 35, 383imtr4i 190 . . 3 (((A<P B 𝑞 Q) (𝑞 <Q 𝑟 𝑟 (1st𝐶))) → 𝑞 (1st𝐶))
4039ex 108 . 2 ((A<P B 𝑞 Q) → ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) → 𝑞 (1st𝐶)))
4140rexlimdvw 2430 1 ((A<P B 𝑞 Q) → (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) → 𝑞 (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <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-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-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-enq 6331  df-nqqs 6332  df-plqqs 6333  df-ltnqqs 6337  df-inp 6449  df-iltp 6453
This theorem is referenced by:  ltexprlemrnd  6579
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