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Theorem ltexprlemopl 6565
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6577. (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
ltexprlemopl ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
Distinct variable groups:   x,y,𝑞,𝑟,A   x,B,y,𝑞,𝑟   x,𝐶,y,𝑞,𝑟

Proof of Theorem ltexprlemopl
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . 5 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
21ltexprlemell 6562 . . . 4 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
32simprbi 260 . . 3 (𝑞 (1st𝐶) → y(y (2ndA) (y +Q 𝑞) (1stB)))
4 19.42v 1783 . . . . . . . 8 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))))
5 19.42v 1783 . . . . . . . . 9 (y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
65anbi2i 430 . . . . . . . 8 ((A<P B y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
74, 6bitri 173 . . . . . . 7 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) ↔ (A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
8 ltrelpr 6480 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4334 . . . . . . . . . . . . 13 (A<P B → (A P B P))
109simprd 107 . . . . . . . . . . . 12 (A<P BB P)
11 prop 6450 . . . . . . . . . . . . 13 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6463 . . . . . . . . . . . . 13 ((⟨(1stB), (2ndB)⟩ P (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1311, 12sylan 267 . . . . . . . . . . . 12 ((B P (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1410, 13sylan 267 . . . . . . . . . . 11 ((A<P B (y +Q 𝑞) (1stB)) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1514adantrl 447 . . . . . . . . . 10 ((A<P B (y (2ndA) (y +Q 𝑞) (1stB))) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1615adantrl 447 . . . . . . . . 9 ((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
179simpld 105 . . . . . . . . . . . . . . 15 (A<P BA P)
1817ad2antrr 457 . . . . . . . . . . . . . 14 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → A P)
19 simplrr 488 . . . . . . . . . . . . . . 15 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y (2ndA) (y +Q 𝑞) (1stB)))
2019simpld 105 . . . . . . . . . . . . . 14 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y (2ndA))
21 prop 6450 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
22 elprnqu 6457 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
2321, 22sylan 267 . . . . . . . . . . . . . 14 ((A P y (2ndA)) → y Q)
2418, 20, 23syl2anc 391 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y Q)
25 simplrl 487 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑞 Q)
26 ltaddnq 6383 . . . . . . . . . . . . 13 ((y Q 𝑞 Q) → y <Q (y +Q 𝑞))
2724, 25, 26syl2anc 391 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y <Q (y +Q 𝑞))
28 simprr 484 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y +Q 𝑞) <Q 𝑠)
29 ltsonq 6375 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6342 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4662 . . . . . . . . . . . 12 ((y <Q (y +Q 𝑞) (y +Q 𝑞) <Q 𝑠) → y <Q 𝑠)
3227, 28, 31syl2anc 391 . . . . . . . . . . 11 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y <Q 𝑠)
3310ad2antrr 457 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → B P)
34 simprl 483 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 (1stB))
35 elprnql 6456 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P 𝑠 (1stB)) → 𝑠 Q)
3611, 35sylan 267 . . . . . . . . . . . . 13 ((B P 𝑠 (1stB)) → 𝑠 Q)
3733, 34, 36syl2anc 391 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 Q)
38 ltexnqq 6384 . . . . . . . . . . . 12 ((y Q 𝑠 Q) → (y <Q 𝑠𝑟 Q (y +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 391 . . . . . . . . . . 11 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (y <Q 𝑠𝑟 Q (y +Q 𝑟) = 𝑠))
4032, 39mpbid 135 . . . . . . . . . 10 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑟 Q (y +Q 𝑟) = 𝑠)
41 simplrr 488 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑞) <Q 𝑠)
42 simprr 484 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑟) = 𝑠)
4341, 42breqtrrd 3780 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑞) <Q (y +Q 𝑟))
4425adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑞 Q)
45 simprl 483 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑟 Q)
4624adantr 261 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → y Q)
47 ltanqg 6377 . . . . . . . . . . . . . . 15 ((𝑞 Q 𝑟 Q y Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
4844, 45, 46, 47syl3anc 1134 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
4943, 48mpbird 156 . . . . . . . . . . . . 13 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑞 <Q 𝑟)
5020adantr 261 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → y (2ndA))
51 simplrl 487 . . . . . . . . . . . . . . 15 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → 𝑠 (1stB))
5242, 51eqeltrd 2111 . . . . . . . . . . . . . 14 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y +Q 𝑟) (1stB))
5350, 52jca 290 . . . . . . . . . . . . 13 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (y (2ndA) (y +Q 𝑟) (1stB)))
5449, 45, 53jca32 293 . . . . . . . . . . . 12 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) (𝑟 Q (y +Q 𝑟) = 𝑠)) → (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5554expr 357 . . . . . . . . . . 11 ((((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) 𝑟 Q) → ((y +Q 𝑟) = 𝑠 → (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
5655reximdva 2415 . . . . . . . . . 10 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → (𝑟 Q (y +Q 𝑟) = 𝑠𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))))))
5740, 56mpd 13 . . . . . . . . 9 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5816, 57rexlimddv 2431 . . . . . . . 8 ((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
5958eximi 1488 . . . . . . 7 (y(A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) → y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
607, 59sylbir 125 . . . . . 6 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
61 rexcom4 2571 . . . . . 6 (𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ y𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
6260, 61sylibr 137 . . . . 5 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))))
63 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)))))
64 19.42v 1783 . . . . . . . 8 (y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
6564anbi2i 430 . . . . . . 7 ((𝑞 <Q 𝑟 y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6663, 65bitri 173 . . . . . 6 (y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6766rexbii 2325 . . . . 5 (𝑟 Q y(𝑞 <Q 𝑟 (𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB)))) ↔ 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
6862, 67sylib 127 . . . 4 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
691ltexprlemell 6562 . . . . . 6 (𝑟 (1st𝐶) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
7069anbi2i 430 . . . . 5 ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
7170rexbii 2325 . . . 4 (𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ 𝑟 Q (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
7268, 71sylibr 137 . . 3 ((A<P B (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
733, 72sylanr2 385 . 2 ((A<P B (𝑞 Q 𝑞 (1st𝐶))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
74733impb 1099 1 ((A<P B 𝑞 Q 𝑞 (1st𝐶)) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
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  cop 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259   <Q cltq 6262  Pcnp 6268  <P cltp 6272
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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-ltnqqs 6330  df-inp 6441  df-iltp 6445
This theorem is referenced by:  ltexprlemrnd  6569
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