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Theorem ltexprlemdisj 6580
Description: Our constructed difference is disjoint. 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
ltexprlemdisj 
<P  q  Q.  q  1st `  C  q  2nd `  C
Distinct variable groups:   ,, q,   ,,, q   , C,, q

Proof of Theorem ltexprlemdisj
Dummy variables  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 6382 . . . . . 6  <Q  Or  Q.
2 ltrelnq 6349 . . . . . 6  <Q  C_  Q.  X.  Q.
31, 2son2lpi 4664 . . . . 5  <Q  <Q
4 ltrelpr 6488 . . . . . . . . . . . . . . . 16  <P  C_  P.  X.  P.
54brel 4335 . . . . . . . . . . . . . . 15 
<P  P.  P.
65simprd 107 . . . . . . . . . . . . . 14 
<P  P.
7 prop 6458 . . . . . . . . . . . . . 14  P.  <. 1st `  ,  2nd `  >.  P.
86, 7syl 14 . . . . . . . . . . . . 13 
<P  <. 1st `  ,  2nd `  >.  P.
9 prltlu 6470 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  +Q  q  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
108, 9syl3an1 1167 . . . . . . . . . . . 12  <P  +Q  q  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
11103expb 1104 . . . . . . . . . . 11  <P  +Q  q  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
1211adantlr 446 . . . . . . . . . 10  <P  q  Q.  +Q  q  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
1312adantrll 453 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
1413adantrrl 455 . . . . . . . 8  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  q
15 ltanqg 6384 . . . . . . . . . 10  Q.  Q.  h  Q.  <Q  h  +Q  <Q  h  +Q
1615adantl 262 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  Q.  Q.  h 
Q.  <Q  h  +Q  <Q  h  +Q
175simpld 105 . . . . . . . . . . . . 13 
<P  P.
18 prop 6458 . . . . . . . . . . . . 13  P.  <. 1st `  ,  2nd `  >.  P.
1917, 18syl 14 . . . . . . . . . . . 12 
<P  <. 1st `  ,  2nd `  >.  P.
20 elprnqu 6465 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
2119, 20sylan 267 . . . . . . . . . . 11  <P  2nd `  Q.
2221ad2ant2r 478 . . . . . . . . . 10  <P  q  Q.  2nd `  +Q  q  1st `  Q.
2322adantrr 448 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd ` 
Q.
24 elprnql 6464 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
2519, 24sylan 267 . . . . . . . . . . 11  <P  1st `  Q.
2625ad2ant2r 478 . . . . . . . . . 10  <P  q  Q.  1st `  +Q  q  2nd `  Q.
2726adantrl 447 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd ` 
Q.
28 simplr 482 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  q 
Q.
29 addcomnqg 6365 . . . . . . . . . 10  Q.  Q.  +Q  +Q
3029adantl 262 . . . . . . . . 9  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  Q.  Q.  +Q  +Q
3116, 23, 27, 28, 30caovord2d 5612 . . . . . . . 8  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd ` 
<Q  +Q  q  <Q  +Q  q
3214, 31mpbird 156 . . . . . . 7  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  <Q
33 prltlu 6470 . . . . . . . . . . . . 13 
<. 1st `  ,  2nd `  >.  P.  1st `  2nd `  <Q
3419, 33syl3an1 1167 . . . . . . . . . . . 12  <P  1st `  2nd `  <Q
35343com23 1109 . . . . . . . . . . 11  <P  2nd `  1st `  <Q
36353expb 1104 . . . . . . . . . 10  <P  2nd `  1st `  <Q
3736adantlr 446 . . . . . . . . 9  <P  q  Q.  2nd `  1st `  <Q
3837adantrlr 454 . . . . . . . 8  <P  q  Q.  2nd `  +Q  q  1st `  1st `  <Q
3938adantrrr 456 . . . . . . 7  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  <Q
4032, 39jca 290 . . . . . 6  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd ` 
<Q  <Q
4140ex 108 . . . . 5  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  <Q  <Q
423, 41mtoi 589 . . . 4  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
4342alrimivv 1752 . . 3  <P  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
44 ltexprlem.1 . . . . . . . . . . . 12  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
4544ltexprlemell 6572 . . . . . . . . . . 11  q  1st `  C  q  Q.  2nd `  +Q  q  1st `
4644ltexprlemelu 6573 . . . . . . . . . . 11  q  2nd `  C  q  Q.  1st `  +Q  q  2nd `
4745, 46anbi12i 433 . . . . . . . . . 10  q  1st `  C  q  2nd `  C 
q  Q.  2nd `  +Q  q  1st `  q  Q.  1st `  +Q  q  2nd `
48 anandi 524 . . . . . . . . . 10  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  q  Q.  2nd `  +Q  q  1st `  q  Q.  1st `  +Q  q  2nd `
4947, 48bitr4i 176 . . . . . . . . 9  q  1st `  C  q  2nd `  C  q  Q.  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
5049baib 827 . . . . . . . 8  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
51 eleq1 2097 . . . . . . . . . . 11  1st `  1st `
52 oveq1 5462 . . . . . . . . . . . 12  +Q  q  +Q  q
5352eleq1d 2103 . . . . . . . . . . 11  +Q  q  2nd `  +Q  q  2nd `
5451, 53anbi12d 442 . . . . . . . . . 10  1st `  +Q  q  2nd `  1st `  +Q  q  2nd `
5554cbvexv 1792 . . . . . . . . 9  1st `  +Q  q  2nd `  1st `  +Q  q  2nd `
5655anbi2i 430 . . . . . . . 8  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
5750, 56syl6bb 185 . . . . . . 7  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
58 eeanv 1804 . . . . . . 7  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
5957, 58syl6bbr 187 . . . . . 6  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6059notbid 591 . . . . 5  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
61 alnex 1385 . . . . . . 7  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6261albii 1356 . . . . . 6  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
63 alnex 1385 . . . . . 6  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6462, 63bitri 173 . . . . 5  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6560, 64syl6bbr 187 . . . 4  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6665adantl 262 . . 3  <P  q  Q.  q  1st `  C  q  2nd `  C  2nd `  +Q  q  1st `  1st `  +Q  q  2nd `
6743, 66mpbird 156 . 2  <P  q  Q.  q  1st `  C  q  2nd `  C
6867ralrimiva 2386 1 
<P  q  Q.  q  1st `  C  q  2nd `  C
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3a 884  wal 1240   wceq 1242  wex 1378   wcel 1390  wral 2300   {crab 2304   <.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-po 4024  df-iso 4025  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-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-enq 6331  df-nqqs 6332  df-plqqs 6333  df-ltnqqs 6337  df-inp 6449  df-iltp 6453
This theorem is referenced by:  ltexprlempr  6582
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