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Theorem ltexprlemupu 6578
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 6587. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
Assertion
Ref Expression
ltexprlemupu  <P  r  Q.  q 
Q.  q  <Q 
r  q  2nd `  C  r  2nd `  C
Distinct variable groups:   ,, q, r,   ,,, q, r   , C,, q, r

Proof of Theorem ltexprlemupu
StepHypRef Expression
1 simplr 482 . . . . . 6  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  r  Q.
2 simprrr 492 . . . . . . 7  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  1st `  +Q  q  2nd `
32simpld 105 . . . . . 6  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  1st `
4 simprl 483 . . . . . . . 8  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q  <Q  r
5 simpll 481 . . . . . . . . 9  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  <P
6 simprrl 491 . . . . . . . . . 10  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  1st `
76adantl 262 . . . . . . . . 9  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  1st `
8 ltrelpr 6488 . . . . . . . . . . . . 13  <P  C_  P.  X.  P.
98brel 4335 . . . . . . . . . . . 12 
<P  P.  P.
109simpld 105 . . . . . . . . . . 11 
<P  P.
11 prop 6458 . . . . . . . . . . 11  P.  <. 1st `  ,  2nd `  >.  P.
1210, 11syl 14 . . . . . . . . . 10 
<P  <. 1st `  ,  2nd `  >.  P.
13 elprnql 6464 . . . . . . . . . 10 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
1412, 13sylan 267 . . . . . . . . 9  <P  1st `  Q.
155, 7, 14syl2anc 391 . . . . . . . 8  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  Q.
16 ltanqi 6386 . . . . . . . 8  q  <Q  r  Q.  +Q  q  <Q  +Q  r
174, 15, 16syl2anc 391 . . . . . . 7  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  r
189simprd 107 . . . . . . . . 9 
<P  P.
195, 18syl 14 . . . . . . . 8  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  P.
202simprd 107 . . . . . . . 8  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  +Q  q  2nd `
21 prop 6458 . . . . . . . . 9  P.  <. 1st `  ,  2nd `  >.  P.
22 prcunqu 6468 . . . . . . . . 9 
<. 1st `  ,  2nd `  >.  P.  +Q  q  2nd `  +Q  q  <Q  +Q  r  +Q  r  2nd `
2321, 22sylan 267 . . . . . . . 8  P.  +Q  q  2nd `  +Q  q  <Q  +Q  r  +Q  r  2nd `
2419, 20, 23syl2anc 391 . . . . . . 7  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  +Q  q  <Q  +Q  r  +Q  r  2nd `
2517, 24mpd 13 . . . . . 6  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  +Q  r  2nd `
261, 3, 25jca32 293 . . . . 5  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd ` 
r  Q.  1st `  +Q  r  2nd `
2726eximi 1488 . . . 4  <P  r  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `  r 
Q.  1st `  +Q  r  2nd `
28 ltexprlem.1 . . . . . . . . . 10  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
2928ltexprlemelu 6573 . . . . . . . . 9  q  2nd `  C  q  Q.  1st `  +Q  q  2nd `
30 19.42v 1783 . . . . . . . . 9  q  Q.  1st `  +Q  q  2nd `  q  Q.  1st `  +Q  q  2nd `
3129, 30bitr4i 176 . . . . . . . 8  q  2nd `  C  q  Q.  1st `  +Q  q  2nd `
3231anbi2i 430 . . . . . . 7  q  <Q  r  q  2nd `  C  q 
<Q  r  q  Q.  1st `  +Q  q  2nd `
33 19.42v 1783 . . . . . . 7  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
3432, 33bitr4i 176 . . . . . 6  q  <Q  r  q  2nd `  C  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
3534anbi2i 430 . . . . 5  <P  r  Q.  q  <Q  r  q  2nd `  C 
<P  r 
Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
36 19.42v 1783 . . . . 5  <P  r  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `  <P  r  Q.  q 
<Q  r 
q  Q.  1st `  +Q  q  2nd `
3735, 36bitr4i 176 . . . 4  <P  r  Q.  q  <Q  r  q  2nd `  C  <P  r  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
3828ltexprlemelu 6573 . . . . 5  r  2nd `  C  r  Q.  1st `  +Q  r  2nd `
39 19.42v 1783 . . . . 5  r  Q.  1st `  +Q  r  2nd `  r  Q.  1st `  +Q  r  2nd `
4038, 39bitr4i 176 . . . 4  r  2nd `  C  r  Q.  1st `  +Q  r  2nd `
4127, 37, 403imtr4i 190 . . 3  <P  r  Q.  q  <Q  r  q  2nd `  C  r  2nd `  C
4241ex 108 . 2  <P  r  Q.  q  <Q 
r  q  2nd `  C  r  2nd `  C
4342rexlimdvw 2430 1  <P  r  Q.  q 
Q.  q  <Q 
r  q  2nd `  C  r  2nd `  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390  wrex 2301   {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-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:  ltexprlemrnd  6579
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