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Theorem recexprlempr 6604
Description: is a positive real. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
Assertion
Ref Expression
recexprlempr  P.  P.
Distinct variable groups:   ,,   ,,

Proof of Theorem recexprlempr
Dummy variables  r  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recexpr.1 . . . 4  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
21recexprlemm 6596 . . 3  P.  q  Q.  q  1st `  r  Q.  r  2nd `
3 ltrelnq 6349 . . . . . . . . . . 11  <Q  C_  Q.  X.  Q.
43brel 4335 . . . . . . . . . 10 
<Q  Q.  Q.
54simpld 105 . . . . . . . . 9 
<Q  Q.
65adantr 261 . . . . . . . 8  <Q  *Q `  2nd `  Q.
76exlimiv 1486 . . . . . . 7  <Q  *Q `  2nd `  Q.
87abssi 3009 . . . . . 6  {  | 
<Q  *Q `  2nd `  }  C_  Q.
9 nqex 6347 . . . . . . 7  Q.  _V
109elpw2 3902 . . . . . 6  {  |  <Q  *Q `  2nd `  }  ~P Q.  {  |  <Q  *Q `  2nd `  }  C_ 
Q.
118, 10mpbir 134 . . . . 5  {  | 
<Q  *Q `  2nd `  }  ~P Q.
123brel 4335 . . . . . . . . . 10 
<Q  Q.  Q.
1312simprd 107 . . . . . . . . 9 
<Q  Q.
1413adantr 261 . . . . . . . 8  <Q  *Q `  1st `  Q.
1514exlimiv 1486 . . . . . . 7  <Q  *Q `  1st `  Q.
1615abssi 3009 . . . . . 6  {  | 
<Q  *Q `  1st `  }  C_  Q.
179elpw2 3902 . . . . . 6  {  |  <Q  *Q `  1st `  }  ~P Q.  {  |  <Q  *Q `  1st `  }  C_ 
Q.
1816, 17mpbir 134 . . . . 5  {  | 
<Q  *Q `  1st `  }  ~P Q.
19 opelxpi 4319 . . . . 5  {  |  <Q  *Q `  2nd `  }  ~P Q.  {  |  <Q  *Q `  1st `  }  ~P Q.  <. {  | 
<Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.  ~P Q.  X.  ~P Q.
2011, 18, 19mp2an 402 . . . 4  <. {  | 
<Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.  ~P Q.  X.  ~P Q.
211, 20eqeltri 2107 . . 3  ~P Q.  X.  ~P Q.
222, 21jctil 295 . 2  P.  ~P Q.  X.  ~P Q.  q 
Q.  q  1st `  r  Q.  r  2nd `
231recexprlemrnd 6601 . . 3  P.  q  Q.  q  1st `  r  Q.  q  <Q 
r  r  1st `  r  Q. 
r  2nd `  q  Q.  q  <Q 
r  q  2nd `
241recexprlemdisj 6602 . . 3  P.  q  Q. 
q  1st `  q  2nd `
251recexprlemloc 6603 . . 3  P.  q  Q.  r  Q.  q  <Q 
r  q  1st `  r  2nd `
2623, 24, 253jca 1083 . 2  P.  q  Q.  q  1st `  r  Q.  q  <Q 
r  r  1st `  r  Q. 
r  2nd `  q  Q.  q  <Q 
r  q  2nd `  q  Q.  q  1st `  q  2nd `  q  Q.  r  Q. 
q  <Q  r 
q  1st `  r  2nd `
27 elnp1st2nd 6459 . 2  P.  ~P Q.  X.  ~P Q.  q  Q.  q  1st `  r  Q.  r  2nd `  q  Q.  q  1st `  r  Q.  q  <Q  r  r  1st `  r  Q.  r  2nd `  q  Q.  q  <Q 
r  q  2nd `  q  Q.  q  1st `  q  2nd `  q  Q.  r  Q. 
q  <Q  r 
q  1st `  r  2nd `
2822, 26, 27sylanbrc 394 1  P.  P.
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   w3a 884   wceq 1242  wex 1378   wcel 1390   {cab 2023  wral 2300  wrex 2301    C_ wss 2911   ~Pcpw 3351   <.cop 3370   class class class wbr 3755    X. cxp 4286   ` cfv 4845   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   *Qcrq 6268    <Q cltq 6269   P.cnp 6275
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-rq 6336  df-ltnqqs 6337  df-inp 6449
This theorem is referenced by:  recexprlem1ssl  6605  recexprlem1ssu  6606  recexprlemss1l  6607  recexprlemss1u  6608  recexprlemex  6609  recexpr  6610
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