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Theorem ltexprlemm 6574
Description: Our constructed difference is inhabited. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
Assertion
Ref Expression
ltexprlemm 
<P  q  Q.  q  1st `  C  r 
Q.  r  2nd `  C
Distinct variable groups:   ,, q, r,   ,,, q, r   , C,, q, r

Proof of Theorem ltexprlemm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6488 . . . . . . . . 9  <P  C_  P.  X.  P.
21brel 4335 . . . . . . . 8 
<P  P.  P.
3 ltdfpr 6489 . . . . . . . . 9  P.  P.  <P  Q.  2nd `  1st `
43biimpd 132 . . . . . . . 8  P.  P.  <P  Q.  2nd `  1st `
52, 4mpcom 32 . . . . . . 7 
<P  Q.  2nd `  1st `
6 simprrl 491 . . . . . . . . . 10  <P  Q.  2nd `  1st `  2nd `
72simprd 107 . . . . . . . . . . . . 13 
<P  P.
8 prop 6458 . . . . . . . . . . . . . . . . . 18  P.  <. 1st `  ,  2nd `  >.  P.
9 prnmaxl 6471 . . . . . . . . . . . . . . . . . 18 
<. 1st `  ,  2nd `  >.  P.  1st `  1st ` 
<Q
108, 9sylan 267 . . . . . . . . . . . . . . . . 17  P.  1st `  1st ` 
<Q
11 ltexnqi 6392 . . . . . . . . . . . . . . . . . 18 
<Q  q  Q.  +Q  q
1211reximi 2410 . . . . . . . . . . . . . . . . 17  1st `  <Q  1st `  q  Q.  +Q  q
1310, 12syl 14 . . . . . . . . . . . . . . . 16  P.  1st `  1st `  q  Q.  +Q  q
14 df-rex 2306 . . . . . . . . . . . . . . . 16  1st `  q  Q.  +Q  q  1st `  q 
Q.  +Q  q
1513, 14sylib 127 . . . . . . . . . . . . . . 15  P.  1st `  1st `  q 
Q.  +Q  q
16 r19.42v 2461 . . . . . . . . . . . . . . . 16  q  Q.  1st `  +Q  q  1st `  q 
Q.  +Q  q
1716exbii 1493 . . . . . . . . . . . . . . 15  q  Q.  1st `  +Q  q  1st `  q 
Q.  +Q  q
1815, 17sylibr 137 . . . . . . . . . . . . . 14  P.  1st `  q 
Q.  1st `  +Q  q
19 eleq1 2097 . . . . . . . . . . . . . . . . 17  +Q  q  +Q  q  1st `  1st `
2019biimparc 283 . . . . . . . . . . . . . . . 16  1st `  +Q  q  +Q  q  1st `
2120reximi 2410 . . . . . . . . . . . . . . 15  q  Q.  1st `  +Q  q  q  Q.  +Q  q  1st `
2221exlimiv 1486 . . . . . . . . . . . . . 14  q  Q.  1st `  +Q  q  q  Q.  +Q  q  1st `
2318, 22syl 14 . . . . . . . . . . . . 13  P.  1st `  q  Q.  +Q  q  1st `
247, 23sylan 267 . . . . . . . . . . . 12  <P  1st `  q  Q.  +Q  q  1st `
2524adantrl 447 . . . . . . . . . . 11  <P  2nd `  1st `  q  Q.  +Q  q  1st `
2625adantrl 447 . . . . . . . . . 10  <P  Q.  2nd `  1st `  q  Q.  +Q  q  1st `
276, 26jca 290 . . . . . . . . 9  <P  Q.  2nd `  1st `  2nd `  q  Q.  +Q  q  1st `
2827expr 357 . . . . . . . 8  <P  Q.  2nd `  1st `  2nd `  q 
Q.  +Q  q  1st `
2928reximdva 2415 . . . . . . 7 
<P  Q.  2nd `  1st `  Q.  2nd `  q 
Q.  +Q  q  1st `
305, 29mpd 13 . . . . . 6 
<P  Q.  2nd `  q 
Q.  +Q  q  1st `
31 r19.42v 2461 . . . . . . 7  q  Q.  2nd `  +Q  q  1st `  2nd `  q 
Q.  +Q  q  1st `
3231rexbii 2325 . . . . . 6  Q.  q  Q.  2nd `  +Q  q  1st `  Q.  2nd `  q  Q.  +Q  q  1st `
3330, 32sylibr 137 . . . . 5 
<P  Q.  q 
Q.  2nd `  +Q  q  1st `
34 rexcom 2468 . . . . 5  Q.  q  Q.  2nd `  +Q  q  1st `  q  Q.  Q.  2nd `  +Q  q  1st `
3533, 34sylib 127 . . . 4 
<P  q  Q. 
Q.  2nd `  +Q  q  1st `
362simpld 105 . . . . . . . . . . . 12 
<P  P.
37 prop 6458 . . . . . . . . . . . . 13  P.  <. 1st `  ,  2nd `  >.  P.
38 elprnqu 6465 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
3937, 38sylan 267 . . . . . . . . . . . 12  P.  2nd `  Q.
4036, 39sylan 267 . . . . . . . . . . 11  <P  2nd `  Q.
4140ex 108 . . . . . . . . . 10 
<P  2nd `  Q.
4241pm4.71rd 374 . . . . . . . . 9 
<P  2nd `  Q.  2nd `
4342anbi1d 438 . . . . . . . 8 
<P  2nd `  +Q  q  1st `  Q.  2nd `  +Q  q  1st `
44 anass 381 . . . . . . . 8  Q.  2nd `  +Q  q  1st `  Q.  2nd `  +Q  q  1st `
4543, 44syl6bb 185 . . . . . . 7 
<P  2nd `  +Q  q  1st `  Q.  2nd `  +Q  q  1st `
4645exbidv 1703 . . . . . 6 
<P  2nd `  +Q  q  1st `  Q.  2nd `  +Q  q  1st `
4746rexbidv 2321 . . . . 5 
<P  q  Q.  2nd `  +Q  q  1st `  q  Q.  Q.  2nd `  +Q  q  1st `
48 df-rex 2306 . . . . . 6  Q.  2nd `  +Q  q  1st `  Q.  2nd `  +Q  q  1st `
4948rexbii 2325 . . . . 5  q  Q.  Q.  2nd `  +Q  q  1st `  q  Q. 
Q.  2nd `  +Q  q  1st `
5047, 49syl6bbr 187 . . . 4 
<P  q  Q.  2nd `  +Q  q  1st `  q  Q. 
Q.  2nd `  +Q  q  1st `
5135, 50mpbird 156 . . 3 
<P  q  Q.  2nd `  +Q  q  1st `
52 ltexprlem.1 . . . . . 6  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
5352ltexprlemell 6572 . . . . 5  q  1st `  C  q  Q.  2nd `  +Q  q  1st `
5453rexbii 2325 . . . 4  q  Q.  q  1st `  C  q  Q.  q  Q.  2nd `  +Q  q  1st `
55 ssid 2958 . . . . 5  Q.  C_  Q.
56 rexss 3001 . . . . 5  Q.  C_  Q.  q  Q.  2nd `  +Q  q  1st `  q  Q.  q 
Q.  2nd `  +Q  q  1st `
5755, 56ax-mp 7 . . . 4  q  Q.  2nd `  +Q  q  1st `  q  Q.  q 
Q.  2nd `  +Q  q  1st `
5854, 57bitr4i 176 . . 3  q  Q.  q  1st `  C  q  Q.  2nd `  +Q  q  1st `
5951, 58sylibr 137 . 2 
<P  q  Q.  q  1st `  C
60 nfv 1418 . . 3  F/ r  <P
61 nfre1 2359 . . 3  F/ r r  Q.  r  2nd `  C
62 prmu 6461 . . . . 5  <. 1st `  ,  2nd `  >.  P.  r  Q.  r  2nd `
63 rexex 2362 . . . . 5  r  Q.  r  2nd `  r  r  2nd `
6462, 63syl 14 . . . 4  <. 1st `  ,  2nd `  >.  P.  r  r  2nd `
657, 8, 643syl 17 . . 3 
<P  r 
r  2nd `
66 elprnqu 6465 . . . . . . 7 
<. 1st `  ,  2nd `  >.  P.  r  2nd `  r  Q.
678, 66sylan 267 . . . . . 6  P.  r  2nd `  r  Q.
687, 67sylan 267 . . . . 5  <P  r  2nd `  r  Q.
69 prml 6460 . . . . . . . . 9  <. 1st `  ,  2nd `  >.  P.  Q.  1st `
7037, 69syl 14 . . . . . . . 8  P.  Q.  1st `
71 rexex 2362 . . . . . . . 8  Q.  1st `  1st `
7236, 70, 713syl 17 . . . . . . 7 
<P  1st `
7372adantr 261 . . . . . 6  <P  r  2nd `  1st `
74683adant3 923 . . . . . . . . 9  <P  r  2nd `  1st `  r  Q.
75 simp3 905 . . . . . . . . . 10  <P  r  2nd `  1st `  1st `
76 elprnql 6464 . . . . . . . . . . . . . . 15 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
7737, 76sylan 267 . . . . . . . . . . . . . 14  P.  1st `  Q.
7836, 77sylan 267 . . . . . . . . . . . . 13  <P  1st `  Q.
79783adant2 922 . . . . . . . . . . . 12  <P  r  2nd `  1st `  Q.
80 addcomnqg 6365 . . . . . . . . . . . 12  r  Q.  Q.  r  +Q  +Q  r
8174, 79, 80syl2anc 391 . . . . . . . . . . 11  <P  r  2nd `  1st ` 
r  +Q  +Q  r
82 ltaddnq 6390 . . . . . . . . . . . . 13  r  Q.  Q.  r  <Q  r  +Q
8374, 79, 82syl2anc 391 . . . . . . . . . . . 12  <P  r  2nd `  1st `  r  <Q  r  +Q
84 prcunqu 6468 . . . . . . . . . . . . . . 15 
<. 1st `  ,  2nd `  >.  P.  r  2nd `  r  <Q  r  +Q  r  +Q  2nd `
858, 84sylan 267 . . . . . . . . . . . . . 14  P.  r  2nd `  r  <Q  r  +Q  r  +Q  2nd `
867, 85sylan 267 . . . . . . . . . . . . 13  <P  r  2nd `  r  <Q  r  +Q  r  +Q  2nd `
87863adant3 923 . . . . . . . . . . . 12  <P  r  2nd `  1st ` 
r  <Q  r  +Q 
r  +Q  2nd `
8883, 87mpd 13 . . . . . . . . . . 11  <P  r  2nd `  1st ` 
r  +Q  2nd `
8981, 88eqeltrrd 2112 . . . . . . . . . 10  <P  r  2nd `  1st `  +Q  r  2nd `
90 19.8a 1479 . . . . . . . . . 10  1st `  +Q  r  2nd `  1st `  +Q  r  2nd `
9175, 89, 90syl2anc 391 . . . . . . . . 9  <P  r  2nd `  1st `  1st `  +Q  r  2nd `
9274, 91jca 290 . . . . . . . 8  <P  r  2nd `  1st ` 
r  Q.  1st `  +Q  r  2nd `
9352ltexprlemelu 6573 . . . . . . . 8  r  2nd `  C  r  Q.  1st `  +Q  r  2nd `
9492, 93sylibr 137 . . . . . . 7  <P  r  2nd `  1st `  r  2nd `  C
95943expa 1103 . . . . . 6  <P  r  2nd `  1st `  r  2nd `  C
9673, 95exlimddv 1775 . . . . 5  <P  r  2nd `  r  2nd `  C
97 19.8a 1479 . . . . 5  r  Q.  r  2nd `  C  r r 
Q.  r  2nd `  C
9868, 96, 97syl2anc 391 . . . 4  <P  r  2nd `  r r 
Q.  r  2nd `  C
99 df-rex 2306 . . . 4  r  Q.  r  2nd `  C  r r  Q.  r  2nd `  C
10098, 99sylibr 137 . . 3  <P  r  2nd `  r  Q.  r  2nd `  C
10160, 61, 65, 100exlimdd 1749 . 2 
<P  r  Q.  r  2nd `  C
10259, 101jca 290 1 
<P  q  Q.  q  1st `  C  r 
Q.  r  2nd `  C
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    C_ wss 2911   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    +Q cplq 6266    <Q cltq 6269   P.cnp 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-bndl 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 6449  df-iltp 6453
This theorem is referenced by:  ltexprlempr  6582
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