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Theorem ltexprlemopu 6577
Description: The upper cut of our constructed difference is open. 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
ltexprlemopu  <P  r  Q.  r  2nd `  C  q  Q.  q  <Q 
r  q  2nd `  C
Distinct variable groups:   ,, q, r,   ,,, q, r   , C,, q, r

Proof of Theorem ltexprlemopu
Dummy variables  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . 5  C 
<. {  Q.  |  2nd `  +Q  1st `  } ,  {  Q.  |  1st `  +Q  2nd `  } >.
21ltexprlemelu 6573 . . . 4  r  2nd `  C  r  Q.  1st `  +Q  r  2nd `
32simprbi 260 . . 3  r  2nd `  C  1st `  +Q  r  2nd `
4 19.42v 1783 . . . . . . . 8  <P  r  Q.  1st `  +Q  r  2nd `  <P  r  Q.  1st `  +Q  r  2nd `
5 19.42v 1783 . . . . . . . . 9  r  Q.  1st `  +Q  r  2nd `  r  Q.  1st `  +Q  r  2nd `
65anbi2i 430 . . . . . . . 8  <P  r 
Q.  1st `  +Q  r  2nd ` 
<P  r  Q.  1st `  +Q  r  2nd `
74, 6bitri 173 . . . . . . 7  <P  r  Q.  1st `  +Q  r  2nd `  <P  r  Q.  1st `  +Q  r  2nd `
8 ltrelpr 6488 . . . . . . . . . . . . . . 15  <P  C_  P.  X.  P.
98brel 4335 . . . . . . . . . . . . . 14 
<P  P.  P.
109simprd 107 . . . . . . . . . . . . 13 
<P  P.
11 prop 6458 . . . . . . . . . . . . 13  P.  <. 1st `  ,  2nd `  >.  P.
1210, 11syl 14 . . . . . . . . . . . 12 
<P  <. 1st `  ,  2nd `  >.  P.
13 prnminu 6472 . . . . . . . . . . . 12 
<. 1st `  ,  2nd `  >.  P.  +Q  r  2nd `  s  2nd `  s 
<Q  +Q  r
1412, 13sylan 267 . . . . . . . . . . 11  <P  +Q  r  2nd `  s  2nd `  s 
<Q  +Q  r
1514adantrl 447 . . . . . . . . . 10  <P  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r
1615adantrl 447 . . . . . . . . 9  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r
17 ltdfpr 6489 . . . . . . . . . . . . . . 15  P.  P.  <P  t  Q.  t  2nd `  t  1st `
1817biimpd 132 . . . . . . . . . . . . . 14  P.  P.  <P  t  Q. 
t  2nd `  t  1st `
199, 18mpcom 32 . . . . . . . . . . . . 13 
<P  t  Q.  t  2nd `  t  1st `
2019ad2antrr 457 . . . . . . . . . . . 12  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q.  t  2nd `  t  1st `
219simpld 105 . . . . . . . . . . . . . . . 16 
<P  P.
2221ad2antrr 457 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  P.
2322adantr 261 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  P.
24 simplrr 488 . . . . . . . . . . . . . . . 16  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  1st `  +Q  r  2nd `
2524simpld 105 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  1st `
2625adantr 261 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  1st `
27 simprrl 491 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  t  2nd `
28 prop 6458 . . . . . . . . . . . . . . 15  P.  <. 1st `  ,  2nd `  >.  P.
29 prltlu 6470 . . . . . . . . . . . . . . 15 
<. 1st `  ,  2nd `  >.  P.  1st `  t  2nd `  <Q  t
3028, 29syl3an1 1167 . . . . . . . . . . . . . 14  P.  1st `  t  2nd `  <Q  t
3123, 26, 27, 30syl3anc 1134 . . . . . . . . . . . . 13  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  <Q  t
32 simplll 485 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  <P
33 simprrr 492 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  t  1st `
34 simplrl 487 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  s  2nd `
35 prltlu 6470 . . . . . . . . . . . . . . 15 
<. 1st `  ,  2nd `  >.  P.  t  1st `  s  2nd `  t  <Q  s
3612, 35syl3an1 1167 . . . . . . . . . . . . . 14  <P  t  1st `  s  2nd `  t  <Q  s
3732, 33, 34, 36syl3anc 1134 . . . . . . . . . . . . 13  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  t  <Q  s
38 ltsonq 6382 . . . . . . . . . . . . . 14  <Q  Or  Q.
39 ltrelnq 6349 . . . . . . . . . . . . . 14  <Q  C_  Q.  X.  Q.
4038, 39sotri 4663 . . . . . . . . . . . . 13  <Q  t  t  <Q  s  <Q  s
4131, 37, 40syl2anc 391 . . . . . . . . . . . 12  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  t  Q. 
t  2nd `  t  1st `  <Q  s
4220, 41rexlimddv 2431 . . . . . . . . . . 11  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  <Q  s
43 elprnql 6464 . . . . . . . . . . . . . 14 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
4428, 43sylan 267 . . . . . . . . . . . . 13  P.  1st `  Q.
4522, 25, 44syl2anc 391 . . . . . . . . . . . 12  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r 
Q.
46 elprnqu 6465 . . . . . . . . . . . . . 14 
<. 1st `  ,  2nd `  >.  P.  s  2nd `  s  Q.
4712, 46sylan 267 . . . . . . . . . . . . 13  <P  s  2nd `  s  Q.
4847ad2ant2r 478 . . . . . . . . . . . 12  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  s 
Q.
49 ltexnqq 6391 . . . . . . . . . . . 12  Q.  s  Q.  <Q  s  q  Q.  +Q  q  s
5045, 48, 49syl2anc 391 . . . . . . . . . . 11  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r 
<Q  s  q  Q.  +Q  q  s
5142, 50mpbid 135 . . . . . . . . . 10  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s
52 simprr 484 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  +Q  q  s
53 simplrr 488 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  s  <Q  +Q  r
5452, 53eqbrtrd 3775 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  +Q  q  <Q  +Q  r
55 simprl 483 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  q  Q.
56 simplrl 487 . . . . . . . . . . . . . . . 16  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  r 
Q.
5756adantr 261 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  r  Q.
5845adantr 261 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  Q.
59 ltanqg 6384 . . . . . . . . . . . . . . 15  q  Q.  r  Q.  Q. 
q  <Q  r  +Q  q  <Q  +Q  r
6055, 57, 58, 59syl3anc 1134 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  q  <Q  r  +Q  q  <Q  +Q  r
6154, 60mpbird 156 . . . . . . . . . . . . 13  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  q  <Q  r
6225adantr 261 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  1st `
63 simplrl 487 . . . . . . . . . . . . . . 15  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  s  2nd `
6452, 63eqeltrd 2111 . . . . . . . . . . . . . 14  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  +Q  q  2nd `
6562, 64jca 290 . . . . . . . . . . . . 13  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  1st `  +Q  q  2nd `
6661, 55, 65jca32 293 . . . . . . . . . . . 12  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
6766expr 357 . . . . . . . . . . 11  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q 
Q.  +Q  q  s  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
6867reximdva 2415 . . . . . . . . . 10  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  +Q  q  s  q  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
6951, 68mpd 13 . . . . . . . . 9  <P  r  Q.  1st `  +Q  r  2nd `  s  2nd `  s  <Q  +Q  r  q  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
7016, 69rexlimddv 2431 . . . . . . . 8  <P  r  Q.  1st `  +Q  r  2nd `  q  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
7170eximi 1488 . . . . . . 7  <P  r  Q.  1st `  +Q  r  2nd `  q  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
727, 71sylbir 125 . . . . . 6  <P  r  Q.  1st `  +Q  r  2nd `  q  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
73 rexcom4 2571 . . . . . 6  q  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q  Q. 
q  <Q  r 
q  Q.  1st `  +Q  q  2nd `
7472, 73sylibr 137 . . . . 5  <P  r  Q.  1st `  +Q  r  2nd `  q  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `
75 19.42v 1783 . . . . . . 7  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
76 19.42v 1783 . . . . . . . 8  q  Q.  1st `  +Q  q  2nd `  q  Q.  1st `  +Q  q  2nd `
7776anbi2i 430 . . . . . . 7  q  <Q  r  q 
Q.  1st `  +Q  q  2nd `  q 
<Q  r 
q  Q.  1st `  +Q  q  2nd `
7875, 77bitri 173 . . . . . 6  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
7978rexbii 2325 . . . . 5  q  Q.  q  <Q  r  q  Q.  1st `  +Q  q  2nd `  q 
Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
8074, 79sylib 127 . . . 4  <P  r  Q.  1st `  +Q  r  2nd `  q  Q. 
q  <Q  r 
q  Q.  1st `  +Q  q  2nd `
811ltexprlemelu 6573 . . . . . 6  q  2nd `  C  q  Q.  1st `  +Q  q  2nd `
8281anbi2i 430 . . . . 5  q  <Q  r  q  2nd `  C  q 
<Q  r 
q  Q.  1st `  +Q  q  2nd `
8382rexbii 2325 . . . 4  q  Q. 
q  <Q  r  q  2nd `  C  q  Q.  q  <Q 
r  q  Q.  1st `  +Q  q  2nd `
8480, 83sylibr 137 . . 3  <P  r  Q.  1st `  +Q  r  2nd `  q  Q. 
q  <Q  r  q  2nd `  C
853, 84sylanr2 385 . 2  <P  r  Q.  r  2nd `  C  q  Q.  q  <Q  r  q  2nd `  C
86853impb 1099 1  <P  r  Q.  r  2nd `  C  q  Q.  q  <Q 
r  q  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   <.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-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:  ltexprlemrnd  6579
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