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Theorem ltexprlemm 6431
Description: Our constructed difference is inhabited. Lemma for ltexpri 6444. (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 6353 . . . . . . . . 9 <P ⊆ (P × P)
21brel 4315 . . . . . . . 8 (A<P B → (A P B P))
3 ltdfpr 6354 . . . . . . . . 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 479 . . . . . . . . . 10 ((A<P B (y Q (y (2ndA) y (1stB)))) → y (2ndA))
72simprd 107 . . . . . . . . . . . . 13 (A<P BB P)
8 prop 6323 . . . . . . . . . . . . . . . . . 18 (B P → ⟨(1stB), (2ndB)⟩ P)
9 prnmaxl 6336 . . . . . . . . . . . . . . . . . 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 ltrelnq 6218 . . . . . . . . . . . . . . . . . . . 20 <Q ⊆ (Q × Q)
1211brel 4315 . . . . . . . . . . . . . . . . . . 19 (y <Q w → (y Q w Q))
13 ltexnqq 6260 . . . . . . . . . . . . . . . . . . . 20 ((y Q w Q) → (y <Q w𝑞 Q (y +Q 𝑞) = w))
1413biimpd 132 . . . . . . . . . . . . . . . . . . 19 ((y Q w Q) → (y <Q w𝑞 Q (y +Q 𝑞) = w))
1512, 14mpcom 32 . . . . . . . . . . . . . . . . . 18 (y <Q w𝑞 Q (y +Q 𝑞) = w)
1615reximi 2390 . . . . . . . . . . . . . . . . 17 (w (1stB)y <Q ww (1stB)𝑞 Q (y +Q 𝑞) = w)
1710, 16syl 14 . . . . . . . . . . . . . . . 16 ((B P y (1stB)) → w (1stB)𝑞 Q (y +Q 𝑞) = w)
18 df-rex 2286 . . . . . . . . . . . . . . . 16 (w (1stB)𝑞 Q (y +Q 𝑞) = ww(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
1917, 18sylib 127 . . . . . . . . . . . . . . 15 ((B P y (1stB)) → w(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
20 r19.42v 2441 . . . . . . . . . . . . . . . 16 (𝑞 Q (w (1stB) (y +Q 𝑞) = w) ↔ (w (1stB) 𝑞 Q (y +Q 𝑞) = w))
2120exbii 1474 . . . . . . . . . . . . . . 15 (w𝑞 Q (w (1stB) (y +Q 𝑞) = w) ↔ w(w (1stB) 𝑞 Q (y +Q 𝑞) = w))
2219, 21sylibr 137 . . . . . . . . . . . . . 14 ((B P y (1stB)) → w𝑞 Q (w (1stB) (y +Q 𝑞) = w))
23 eleq1 2078 . . . . . . . . . . . . . . . . 17 ((y +Q 𝑞) = w → ((y +Q 𝑞) (1stB) ↔ w (1stB)))
2423biimparc 283 . . . . . . . . . . . . . . . 16 ((w (1stB) (y +Q 𝑞) = w) → (y +Q 𝑞) (1stB))
2524reximi 2390 . . . . . . . . . . . . . . 15 (𝑞 Q (w (1stB) (y +Q 𝑞) = w) → 𝑞 Q (y +Q 𝑞) (1stB))
2625exlimiv 1467 . . . . . . . . . . . . . 14 (w𝑞 Q (w (1stB) (y +Q 𝑞) = w) → 𝑞 Q (y +Q 𝑞) (1stB))
2722, 26syl 14 . . . . . . . . . . . . 13 ((B P y (1stB)) → 𝑞 Q (y +Q 𝑞) (1stB))
287, 27sylan 267 . . . . . . . . . . . 12 ((A<P B y (1stB)) → 𝑞 Q (y +Q 𝑞) (1stB))
2928adantrl 450 . . . . . . . . . . 11 ((A<P B (y (2ndA) y (1stB))) → 𝑞 Q (y +Q 𝑞) (1stB))
3029adantrl 450 . . . . . . . . . 10 ((A<P B (y Q (y (2ndA) y (1stB)))) → 𝑞 Q (y +Q 𝑞) (1stB))
316, 30jca 290 . . . . . . . . 9 ((A<P B (y Q (y (2ndA) y (1stB)))) → (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
3231expr 357 . . . . . . . 8 ((A<P B y Q) → ((y (2ndA) y (1stB)) → (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB))))
3332reximdva 2395 . . . . . . 7 (A<P B → (y Q (y (2ndA) y (1stB)) → y Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB))))
345, 33mpd 13 . . . . . 6 (A<P By Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
35 r19.42v 2441 . . . . . . 7 (𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
3635rexbii 2305 . . . . . 6 (y Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ y Q (y (2ndA) 𝑞 Q (y +Q 𝑞) (1stB)))
3734, 36sylibr 137 . . . . 5 (A<P By Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)))
38 rexcom 2448 . . . . 5 (y Q 𝑞 Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)))
3937, 38sylib 127 . . . 4 (A<P B𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)))
402simpld 105 . . . . . . . . . . . 12 (A<P BA P)
41 prop 6323 . . . . . . . . . . . . 13 (A P → ⟨(1stA), (2ndA)⟩ P)
42 elprnqu 6330 . . . . . . . . . . . . 13 ((⟨(1stA), (2ndA)⟩ P y (2ndA)) → y Q)
4341, 42sylan 267 . . . . . . . . . . . 12 ((A P y (2ndA)) → y Q)
4440, 43sylan 267 . . . . . . . . . . 11 ((A<P B y (2ndA)) → y Q)
4544ex 108 . . . . . . . . . 10 (A<P B → (y (2ndA) → y Q))
4645pm4.71rd 374 . . . . . . . . 9 (A<P B → (y (2ndA) ↔ (y Q y (2ndA))))
4746anbi1d 441 . . . . . . . 8 (A<P B → ((y (2ndA) (y +Q 𝑞) (1stB)) ↔ ((y Q y (2ndA)) (y +Q 𝑞) (1stB))))
48 anass 383 . . . . . . . 8 (((y Q y (2ndA)) (y +Q 𝑞) (1stB)) ↔ (y Q (y (2ndA) (y +Q 𝑞) (1stB))))
4947, 48syl6bb 185 . . . . . . 7 (A<P B → ((y (2ndA) (y +Q 𝑞) (1stB)) ↔ (y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
5049exbidv 1684 . . . . . 6 (A<P B → (y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ y(y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
5150rexbidv 2301 . . . . 5 (A<P B → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y(y Q (y (2ndA) (y +Q 𝑞) (1stB)))))
52 df-rex 2286 . . . . . 6 (y Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ y(y Q (y (2ndA) (y +Q 𝑞) (1stB))))
5352rexbii 2305 . . . . 5 (𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y(y Q (y (2ndA) (y +Q 𝑞) (1stB))))
5451, 53syl6bbr 187 . . . 4 (A<P B → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q y Q (y (2ndA) (y +Q 𝑞) (1stB))))
5539, 54mpbird 156 . . 3 (A<P B𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))
56 ltexprlem.1 . . . . . 6 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
5756ltexprlemell 6429 . . . . 5 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
5857rexbii 2305 . . . 4 (𝑞 Q 𝑞 (1st𝐶) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
59 ssid 2937 . . . . 5 QQ
60 rexss 2980 . . . . 5 (QQ → (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))))
6159, 60ax-mp 7 . . . 4 (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)) ↔ 𝑞 Q (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
6258, 61bitr4i 176 . . 3 (𝑞 Q 𝑞 (1st𝐶) ↔ 𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB)))
6355, 62sylibr 137 . 2 (A<P B𝑞 Q 𝑞 (1st𝐶))
64 nfv 1398 . . 3 𝑟 A<P B
65 nfre1 2339 . . 3 𝑟𝑟 Q 𝑟 (2nd𝐶)
66 prmu 6326 . . . . 5 (⟨(1stB), (2ndB)⟩ P𝑟 Q 𝑟 (2ndB))
67 rexex 2342 . . . . 5 (𝑟 Q 𝑟 (2ndB) → 𝑟 𝑟 (2ndB))
6866, 67syl 14 . . . 4 (⟨(1stB), (2ndB)⟩ P𝑟 𝑟 (2ndB))
697, 8, 683syl 17 . . 3 (A<P B𝑟 𝑟 (2ndB))
70 elprnqu 6330 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P 𝑟 (2ndB)) → 𝑟 Q)
718, 70sylan 267 . . . . . 6 ((B P 𝑟 (2ndB)) → 𝑟 Q)
727, 71sylan 267 . . . . 5 ((A<P B 𝑟 (2ndB)) → 𝑟 Q)
73 prml 6325 . . . . . . . . 9 (⟨(1stA), (2ndA)⟩ Py Q y (1stA))
7441, 73syl 14 . . . . . . . 8 (A Py Q y (1stA))
75 rexex 2342 . . . . . . . 8 (y Q y (1stA) → y y (1stA))
7640, 74, 753syl 17 . . . . . . 7 (A<P By y (1stA))
7776adantr 261 . . . . . 6 ((A<P B 𝑟 (2ndB)) → y y (1stA))
78723adant3 910 . . . . . . . . 9 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 Q)
79 simp3 892 . . . . . . . . . 10 ((A<P B 𝑟 (2ndB) y (1stA)) → y (1stA))
80 elprnql 6329 . . . . . . . . . . . . . . 15 ((⟨(1stA), (2ndA)⟩ P y (1stA)) → y Q)
8141, 80sylan 267 . . . . . . . . . . . . . 14 ((A P y (1stA)) → y Q)
8240, 81sylan 267 . . . . . . . . . . . . 13 ((A<P B y (1stA)) → y Q)
83823adant2 909 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → y Q)
84 addcomnqg 6234 . . . . . . . . . . . 12 ((𝑟 Q y Q) → (𝑟 +Q y) = (y +Q 𝑟))
8578, 83, 84syl2anc 393 . . . . . . . . . . 11 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 +Q y) = (y +Q 𝑟))
86 ltaddnq 6259 . . . . . . . . . . . . 13 ((𝑟 Q y Q) → 𝑟 <Q (𝑟 +Q y))
8778, 83, 86syl2anc 393 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 <Q (𝑟 +Q y))
88 prcunqu 6333 . . . . . . . . . . . . . . 15 ((⟨(1stB), (2ndB)⟩ P 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
898, 88sylan 267 . . . . . . . . . . . . . 14 ((B P 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
907, 89sylan 267 . . . . . . . . . . . . 13 ((A<P B 𝑟 (2ndB)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
91903adant3 910 . . . . . . . . . . . 12 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 <Q (𝑟 +Q y) → (𝑟 +Q y) (2ndB)))
9287, 91mpd 13 . . . . . . . . . . 11 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 +Q y) (2ndB))
9385, 92eqeltrrd 2093 . . . . . . . . . 10 ((A<P B 𝑟 (2ndB) y (1stA)) → (y +Q 𝑟) (2ndB))
94 19.8a 1460 . . . . . . . . . 10 ((y (1stA) (y +Q 𝑟) (2ndB)) → y(y (1stA) (y +Q 𝑟) (2ndB)))
9579, 93, 94syl2anc 393 . . . . . . . . 9 ((A<P B 𝑟 (2ndB) y (1stA)) → y(y (1stA) (y +Q 𝑟) (2ndB)))
9678, 95jca 290 . . . . . . . 8 ((A<P B 𝑟 (2ndB) y (1stA)) → (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
9756ltexprlemelu 6430 . . . . . . . 8 (𝑟 (2nd𝐶) ↔ (𝑟 Q y(y (1stA) (y +Q 𝑟) (2ndB))))
9896, 97sylibr 137 . . . . . . 7 ((A<P B 𝑟 (2ndB) y (1stA)) → 𝑟 (2nd𝐶))
99983expa 1088 . . . . . 6 (((A<P B 𝑟 (2ndB)) y (1stA)) → 𝑟 (2nd𝐶))
10077, 99exlimddv 1756 . . . . 5 ((A<P B 𝑟 (2ndB)) → 𝑟 (2nd𝐶))
101 19.8a 1460 . . . . 5 ((𝑟 Q 𝑟 (2nd𝐶)) → 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
10272, 100, 101syl2anc 393 . . . 4 ((A<P B 𝑟 (2ndB)) → 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
103 df-rex 2286 . . . 4 (𝑟 Q 𝑟 (2nd𝐶) ↔ 𝑟(𝑟 Q 𝑟 (2nd𝐶)))
104102, 103sylibr 137 . . 3 ((A<P B 𝑟 (2ndB)) → 𝑟 Q 𝑟 (2nd𝐶))
10564, 65, 69, 104exlimdd 1730 . 2 (A<P B𝑟 Q 𝑟 (2nd𝐶))
10663, 105jca 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 871   = wceq 1226  wex 1358   wcel 1370  wrex 2281  {crab 2284  wss 2890  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-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:  ltexprlempr  6439
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