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Theorem ltexprlemm 6573
Description: Our constructed difference is inhabited. Lemma for ltexpri 6586. (Contributed by Jim Kingdon, 17-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
ltexprlemm (A<P B → (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemm
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6487 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4335 . . . . . . . 8 (A<P B → (A P B P))
3 ltdfpr 6488 . . . . . . . . 9 ((A P B P) → (A<P By Q (y (2ndA) y (1stB))))
43biimpd 132 . . . . . . . 8 ((A P B P) → (A<P By Q (y (2ndA) y (1stB))))
52, 4mpcom 32 . . . . . . 7 (A<P By Q (y (2ndA) y (1stB)))
6 simprrl 491 . . . . . . . . . 10 ((A<P B (y Q (y (2ndA) y (1stB)))) → y (2ndA))
72simprd 107 . . . . . . . . . . . . 13 (A<P BB P)
8 prop 6457 . . . . . . . . . . . . . . . . . 18 (B P → ⟨(1stB), (2ndB)⟩ P)
9 prnmaxl 6470 . . . . . . . . . . . . . . . . . 18 ((⟨(1stB), (2ndB)⟩ P y (1stB)) → w (1stB)y <Q w)
108, 9sylan 267 . . . . . . . . . . . . . . . . 17 ((B P y (1stB)) → w (1stB)y <Q w)
11 ltexnqi 6392 . . . . . . . . . . . . . . . . . 18 (y <Q w𝑞 Q (y +Q 𝑞) = w)
1211reximi 2410 . . . . . . . . . . . . . . . . 17 (w (1stB)y <Q ww (1stB)𝑞 Q (y +Q 𝑞) = w)
1310, 12syl 14 . . . . . . . . . . . . . . . 16 ((B P y (1stB)) → w (1stB)𝑞 Q (y +Q 𝑞) = w)
14 df-rex 2306 . . . . . . . . . . . . . . . 16 (w (1stB)𝑞 Q (y +Q 𝑞) = ww(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
1513, 14sylib 127 . . . . . . . . . . . . . . 15 ((B P y (1stB)) → w(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
16 r19.42v 2461 . . . . . . . . . . . . . . . 16 (𝑞 Q (w (1stB) (y +Q 𝑞) = w) ↔ (w (1stB) 𝑞 Q (y +Q 𝑞) = w))
1716exbii 1493 . . . . . . . . . . . . . . 15 (w𝑞 Q (w (1stB) (y +Q 𝑞) = w) ↔ w(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
1815, 17sylibr 137 . . . . . . . . . . . . . 14 ((B P y (1stB)) → w𝑞 Q (w (1stB) (y +Q 𝑞) = w))
19 eleq1 2097 . . . . . . . . . . . . . . . . 17 ((y +Q 𝑞) = w → ((y +Q 𝑞) (1stB) ↔ w (1stB)))
2019biimparc 283 . . . . . . . . . . . . . . . 16 ((w (1stB) (y +Q 𝑞) = w) → (y +Q 𝑞) (1stB))
2120reximi 2410 . . . . . . . . . . . . . . 15 (𝑞 Q (w (1stB) (y +Q 𝑞) = w) → 𝑞 Q (y +Q 𝑞) (1stB))
2221exlimiv 1486 . . . . . . . . . . . . . 14 (w𝑞 Q (w (1stB) (y +Q 𝑞) = w) → 𝑞 Q (y +Q 𝑞) (1stB))
2318, 22syl 14 . . . . . . . . . . . . 13 ((B P y (1stB)) → 𝑞 Q (y +Q 𝑞) (1stB))
247, 23sylan 267 . . . . . . . . . . . 12 ((A<P B y (1stB)) → 𝑞 Q (y +Q 𝑞) (1stB))
2524adantrl 447 . . . . . . . . . . 11 ((A<P B (y (2ndA) y (1stB))) → 𝑞 Q (y +Q 𝑞) (1stB))
2625adantrl 447 . . . . . . . . . 10 ((A<P B (y Q (y (2ndA) y (1stB)))) → 𝑞 Q (y +Q 𝑞) (1stB))
276, 26jca 290 . . . . . . . . 9 ((A<P B (y Q (y (2ndA) y (1stB)))) → (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
2827expr 357 . . . . . . . 8 ((A<P B y Q) → ((y (2ndA) y (1stB)) → (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB))))
2928reximdva 2415 . . . . . . 7 (A<P B → (y Q (y (2ndA) y (1stB)) → y Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB))))
305, 29mpd 13 . . . . . 6 (A<P By Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
31 r19.42v 2461 . . . . . . 7 (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
3231rexbii 2325 . . . . . 6 (y Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ y Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
3330, 32sylibr 137 . . . . 5 (A<P By Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))
34 rexcom 2468 . . . . 5 (y Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)))
3533, 34sylib 127 . . . 4 (A<P B𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)))
362simpld 105 . . . . . . . . . . . 12 (A<P BA P)
37 prop 6457 . . . . . . . . . . . . 13 (A P → ⟨(1stA), (2ndA)⟩ P)
38 elprnqu 6464 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
3937, 38sylan 267 . . . . . . . . . . . 12 ((A P y (2ndA)) → y Q)
4036, 39sylan 267 . . . . . . . . . . 11 ((A<P B y (2ndA)) → y Q)
4140ex 108 . . . . . . . . . 10 (A<P B → (y (2ndA) → y Q))
4241pm4.71rd 374 . . . . . . . . 9 (A<P B → (y (2ndA) ↔ (y Q y (2ndA))))
4342anbi1d 438 . . . . . . . 8 (A<P B → ((y (2ndA) (y +Q 𝑞) (1stB)) ↔ ((y Q y (2ndA)) (y +Q 𝑞) (1stB))))
44 anass 381 . . . . . . . 8 (((y Q y (2ndA)) (y +Q 𝑞) (1stB)) ↔ (y Q (y (2ndA) (y +Q 𝑞) (1stB))))
4543, 44syl6bb 185 . . . . . . 7 (A<P B → ((y (2ndA) (y +Q 𝑞) (1stB)) ↔ (y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
4645exbidv 1703 . . . . . 6 (A<P B → (y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ y(y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
4746rexbidv 2321 . . . . 5 (A<P B → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y(y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
48 df-rex 2306 . . . . . 6 (y Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ y(y Q (y (2ndA) (y +Q 𝑞) (1stB))))
4948rexbii 2325 . . . . 5 (𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y(y Q (y (2ndA) (y +Q 𝑞) (1stB))))
5047, 49syl6bbr 187 . . . 4 (A<P B → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB))))
5135, 50mpbird 156 . . 3 (A<P B𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))
52 ltexprlem.1 . . . . . 6 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
5352ltexprlemell 6571 . . . . 5 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
5453rexbii 2325 . . . 4 (𝑞 Q 𝑞 (1st𝐶) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
55 ssid 2958 . . . . 5 QQ
56 rexss 3001 . . . . 5 (QQ → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
5755, 56ax-mp 7 . . . 4 (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
5854, 57bitr4i 176 . . 3 (𝑞 Q 𝑞 (1st𝐶) ↔ 𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))
5951, 58sylibr 137 . 2 (A<P B𝑞 Q 𝑞 (1st𝐶))
60 nfv 1418 . . 3 𝑟 A<P B
61 nfre1 2359 . . 3 𝑟𝑟 Q 𝑟 (2nd𝐶)
62 prmu 6460 . . . . 5 (⟨(1stB), (2ndB)⟩ P𝑟 Q 𝑟 (2ndB))
63 rexex 2362 . . . . 5 (𝑟 Q 𝑟 (2ndB) → 𝑟 𝑟 (2ndB))
6462, 63syl 14 . . . 4 (⟨(1stB), (2ndB)⟩ P𝑟 𝑟 (2ndB))
657, 8, 643syl 17 . . 3 (A<P B𝑟 𝑟 (2ndB))
66 elprnqu 6464 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P 𝑟 (2ndB)) → 𝑟 Q)
678, 66sylan 267 . . . . . 6 ((B P 𝑟 (2ndB)) → 𝑟 Q)
687, 67sylan 267 . . . . 5 ((A<P B 𝑟 (2ndB)) → 𝑟 Q)
69 prml 6459 . . . . . . . . 9 (⟨(1stA), (2ndA)⟩ Py Q y (1stA))
7037, 69syl 14 . . . . . . . 8 (A Py Q y (1stA))
71 rexex 2362 . . . . . . . 8 (y Q y (1stA) → y y (1stA))
7236, 70, 713syl 17 . . . . . . 7 (A<P By y (1stA))
7372adantr 261 . . . . . 6 ((A<P B 𝑟 (2ndB)) → y y (1stA))
74683adant3 923 . . . . . . . . 9 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 Q)
75 simp3 905 . . . . . . . . . 10 ((A<P B 𝑟 (2ndB) y (1stA)) → y (1stA))
76 elprnql 6463 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
7737, 76sylan 267 . . . . . . . . . . . . . 14 ((A P y (1stA)) → y Q)
7836, 77sylan 267 . . . . . . . . . . . . 13 ((A<P B y (1stA)) → y Q)
79783adant2 922 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → y Q)
80 addcomnqg 6365 . . . . . . . . . . . 12 ((𝑟 Q y Q) → (𝑟 +Q y) = (y +Q 𝑟))
8174, 79, 80syl2anc 391 . . . . . . . . . . 11 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 +Q y) = (y +Q 𝑟))
82 ltaddnq 6390 . . . . . . . . . . . . 13 ((𝑟 Q y Q) → 𝑟 <Q (𝑟 +Q y))
8374, 79, 82syl2anc 391 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 <Q (𝑟 +Q y))
84 prcunqu 6467 . . . . . . . . . . . . . . 15 ((⟨(1stB), (2ndB)⟩ P 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
858, 84sylan 267 . . . . . . . . . . . . . 14 ((B P 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
867, 85sylan 267 . . . . . . . . . . . . 13 ((A<P B 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
87863adant3 923 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
8883, 87mpd 13 . . . . . . . . . . 11 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 +Q y) (2ndB))
8981, 88eqeltrrd 2112 . . . . . . . . . 10 ((A<P B 𝑟 (2ndB) y (1stA)) → (y +Q 𝑟) (2ndB))
90 19.8a 1479 . . . . . . . . . 10 ((y (1stA) (y +Q 𝑟) (2ndB)) → y(y (1stA) (y +Q 𝑟) (2ndB)))
9175, 89, 90syl2anc 391 . . . . . . . . 9 ((A<P B 𝑟 (2ndB) y (1stA)) → y(y (1stA) (y +Q 𝑟) (2ndB)))
9274, 91jca 290 . . . . . . . 8 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
9352ltexprlemelu 6572 . . . . . . . 8 (𝑟 (2nd𝐶) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
9492, 93sylibr 137 . . . . . . 7 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 (2nd𝐶))
95943expa 1103 . . . . . 6 (((A<P B 𝑟 (2ndB)) y (1stA)) → 𝑟 (2nd𝐶))
9673, 95exlimddv 1775 . . . . 5 ((A<P B 𝑟 (2ndB)) → 𝑟 (2nd𝐶))
97 19.8a 1479 . . . . 5 ((𝑟 Q 𝑟 (2nd𝐶)) → 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
9868, 96, 97syl2anc 391 . . . 4 ((A<P B 𝑟 (2ndB)) → 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
99 df-rex 2306 . . . 4 (𝑟 Q 𝑟 (2nd𝐶) ↔ 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
10098, 99sylibr 137 . . 3 ((A<P B 𝑟 (2ndB)) → 𝑟 Q 𝑟 (2nd𝐶))
10160, 61, 65, 100exlimdd 1749 . 2 (A<P B𝑟 Q 𝑟 (2nd𝐶))
10259, 101jca 290 1 (A<P B → (𝑞 Q 𝑞 (1st𝐶) 𝑟 Q 𝑟 (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  wss 2911  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-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-ltnqqs 6337  df-inp 6448  df-iltp 6452
This theorem is referenced by:  ltexprlempr  6581
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