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Theorem ltexprlemopl 6432
Description: The lower cut of our constructed difference is open. Lemma for ltexpri 6444. (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 6429 . . . 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 1764 . . . . . . . 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 1764 . . . . . . . . 9 (y(𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB))) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
65anbi2i 433 . . . . . . . 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 6353 . . . . . . . . . . . . . 14 <P ⊆ (P × P)
98brel 4315 . . . . . . . . . . . . 13 (A<P B → (A P B P))
109simprd 107 . . . . . . . . . . . 12 (A<P BB P)
11 prop 6323 . . . . . . . . . . . . 13 (B P → ⟨(1stB), (2ndB)⟩ P)
12 prnmaxl 6336 . . . . . . . . . . . . 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 450 . . . . . . . . . 10 ((A<P B (y (2ndA) (y +Q 𝑞) (1stB))) → 𝑠 (1stB)(y +Q 𝑞) <Q 𝑠)
1615adantrl 450 . . . . . . . . 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 460 . . . . . . . . . . . . . 14 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → A P)
19 simplrr 476 . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . 15 (A P → ⟨(1stA), (2ndA)⟩ P)
22 elprnqu 6330 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
2321, 22sylan 267 . . . . . . . . . . . . . 14 ((A P y (2ndA)) → y Q)
2418, 20, 23syl2anc 393 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y Q)
25 simplrl 475 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑞 Q)
26 ltaddnq 6259 . . . . . . . . . . . . 13 ((y Q 𝑞 Q) → y <Q (y +Q 𝑞))
2724, 25, 26syl2anc 393 . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 6251 . . . . . . . . . . . . 13 <Q Or Q
30 ltrelnq 6218 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
3129, 30sotri 4643 . . . . . . . . . . . 12 ((y <Q (y +Q 𝑞) (y +Q 𝑞) <Q 𝑠) → y <Q 𝑠)
3227, 28, 31syl2anc 393 . . . . . . . . . . 11 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → y <Q 𝑠)
3310ad2antrr 460 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → B P)
34 simprl 471 . . . . . . . . . . . . 13 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 (1stB))
35 elprnql 6329 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P 𝑠 (1stB)) → 𝑠 Q)
3611, 35sylan 267 . . . . . . . . . . . . 13 ((B P 𝑠 (1stB)) → 𝑠 Q)
3733, 34, 36syl2anc 393 . . . . . . . . . . . 12 (((A<P B (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))) (𝑠 (1stB) (y +Q 𝑞) <Q 𝑠)) → 𝑠 Q)
38 ltexnqq 6260 . . . . . . . . . . . 12 ((y Q 𝑠 Q) → (y <Q 𝑠𝑟 Q (y +Q 𝑟) = 𝑠))
3924, 37, 38syl2anc 393 . . . . . . . . . . 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 476 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 3760 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . 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 6253 . . . . . . . . . . . . . . 15 ((𝑞 Q 𝑟 Q y Q) → (𝑞 <Q 𝑟 ↔ (y +Q 𝑞) <Q (y +Q 𝑟)))
4844, 45, 46, 47syl3anc 1119 . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . 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 2092 . . . . . . . . . . . . . 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 2395 . . . . . . . . . 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 2411 . . . . . . . 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 1469 . . . . . . 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 2550 . . . . . 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 1764 . . . . . . 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 1764 . . . . . . . 8 (y(𝑟 Q (y (2ndA) (y +Q 𝑟) (1stB))) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
6564anbi2i 433 . . . . . . 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 2305 . . . . 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 6429 . . . . . 6 (𝑟 (1st𝐶) ↔ (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB))))
7069anbi2i 433 . . . . 5 ((𝑞 <Q 𝑟 𝑟 (1st𝐶)) ↔ (𝑞 <Q 𝑟 (𝑟 Q y(y (2ndA) (y +Q 𝑟) (1stB)))))
7170rexbii 2305 . . . 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 387 . 2 ((A<P B (𝑞 Q 𝑞 (1st𝐶))) → 𝑟 Q (𝑞 <Q 𝑟 𝑟 (1st𝐶)))
74733impb 1084 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 871   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {crab 2284  cop 3349   class class class wbr 3734  cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136   <Q cltq 6139  Pcnp 6145  <P cltp 6149
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-1o 5912  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-pli 6159  df-mi 6160  df-lti 6161  df-plpq 6197  df-mpq 6198  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-mqqs 6203  df-1nqqs 6204  df-ltnqqs 6206  df-inp 6314  df-iltp 6318
This theorem is referenced by:  ltexprlemrnd  6436
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