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Theorem recexprlemm 6596
Description: is inhabited. 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
recexprlemm  P.  q  Q.  q  1st `  r  Q.  r  2nd `
Distinct variable groups:    r, q,,,   , q,
r,,

Proof of Theorem recexprlemm
StepHypRef Expression
1 prop 6458 . . 3  P.  <. 1st `  ,  2nd `  >.  P.
2 prmu 6461 . . 3  <. 1st `  ,  2nd `  >.  P.  Q.  2nd `
3 recclnq 6376 . . . . . . 7  Q.  *Q ` 
Q.
4 nsmallnqq 6395 . . . . . . 7  *Q `  Q.  q  Q.  q  <Q  *Q `
53, 4syl 14 . . . . . 6  Q.  q  Q.  q  <Q  *Q `
65adantr 261 . . . . 5  Q.  2nd `  q  Q.  q  <Q  *Q `
7 recrecnq 6378 . . . . . . . . . . . 12  Q.  *Q `  *Q `
87eleq1d 2103 . . . . . . . . . . 11  Q.  *Q `  *Q `  2nd `  2nd `
98anbi2d 437 . . . . . . . . . 10  Q.  q  <Q  *Q `  *Q `  *Q `  2nd `  q  <Q  *Q `  2nd `
10 breq2 3759 . . . . . . . . . . . . 13  *Q ` 
q  <Q  q  <Q  *Q `
11 fveq2 5121 . . . . . . . . . . . . . 14  *Q `  *Q `  *Q `  *Q `
1211eleq1d 2103 . . . . . . . . . . . . 13  *Q `  *Q `  2nd `  *Q `  *Q `  2nd `
1310, 12anbi12d 442 . . . . . . . . . . . 12  *Q `  q  <Q  *Q `  2nd `  q  <Q  *Q `  *Q `  *Q `  2nd `
1413spcegv 2635 . . . . . . . . . . 11  *Q `  Q.  q  <Q  *Q `  *Q `  *Q `  2nd `  q  <Q  *Q `  2nd `
153, 14syl 14 . . . . . . . . . 10  Q.  q  <Q  *Q `  *Q `  *Q `  2nd `  q  <Q  *Q `  2nd `
169, 15sylbird 159 . . . . . . . . 9  Q.  q  <Q  *Q `  2nd `  q  <Q  *Q `  2nd `
17 recexpr.1 . . . . . . . . . 10  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
1817recexprlemell 6594 . . . . . . . . 9  q  1st `  q 
<Q  *Q `  2nd `
1916, 18syl6ibr 151 . . . . . . . 8  Q.  q  <Q  *Q `  2nd `  q  1st `
2019expcomd 1327 . . . . . . 7  Q.  2nd ` 
q  <Q  *Q `  q  1st `
2120imp 115 . . . . . 6  Q.  2nd `  q  <Q  *Q `  q  1st `
2221reximdv 2414 . . . . 5  Q.  2nd `  q  Q.  q  <Q  *Q `  q  Q.  q  1st `
236, 22mpd 13 . . . 4  Q.  2nd `  q  Q.  q  1st `
2423rexlimiva 2422 . . 3  Q.  2nd `  q 
Q.  q  1st `
251, 2, 243syl 17 . 2  P.  q  Q.  q  1st `
26 prml 6460 . . 3  <. 1st `  ,  2nd `  >.  P.  Q.  1st `
27 1nq 6350 . . . . . . . 8  1Q  Q.
28 addclnq 6359 . . . . . . . 8  *Q `  Q.  1Q  Q.  *Q `  +Q  1Q 
Q.
293, 27, 28sylancl 392 . . . . . . 7  Q.  *Q `  +Q  1Q 
Q.
30 ltaddnq 6390 . . . . . . . 8  *Q `  Q.  1Q  Q.  *Q `  <Q  *Q `  +Q  1Q
313, 27, 30sylancl 392 . . . . . . 7  Q.  *Q `  <Q  *Q `  +Q  1Q
32 breq2 3759 . . . . . . . 8  r  *Q `  +Q  1Q  *Q `  <Q  r  *Q ` 
<Q  *Q `  +Q  1Q
3332rspcev 2650 . . . . . . 7  *Q `  +Q  1Q  Q.  *Q `  <Q  *Q `  +Q  1Q  r  Q.  *Q ` 
<Q  r
3429, 31, 33syl2anc 391 . . . . . 6  Q.  r  Q.  *Q `  <Q  r
3534adantr 261 . . . . 5  Q.  1st `  r  Q.  *Q ` 
<Q  r
367eleq1d 2103 . . . . . . . . . . 11  Q.  *Q `  *Q `  1st `  1st `
3736anbi2d 437 . . . . . . . . . 10  Q.  *Q `  <Q  r  *Q `  *Q `  1st `  *Q `  <Q  r  1st `
38 breq1 3758 . . . . . . . . . . . . 13  *Q `  <Q  r  *Q `  <Q  r
3911eleq1d 2103 . . . . . . . . . . . . 13  *Q `  *Q `  1st `  *Q `  *Q `  1st `
4038, 39anbi12d 442 . . . . . . . . . . . 12  *Q `  <Q  r  *Q `  1st `  *Q `  <Q  r  *Q `  *Q `  1st `
4140spcegv 2635 . . . . . . . . . . 11  *Q `  Q.  *Q `  <Q  r  *Q `  *Q `  1st `  <Q  r  *Q `  1st `
423, 41syl 14 . . . . . . . . . 10  Q.  *Q `  <Q  r  *Q `  *Q `  1st `  <Q  r  *Q `  1st `
4337, 42sylbird 159 . . . . . . . . 9  Q.  *Q `  <Q  r  1st `  <Q  r  *Q `  1st `
4417recexprlemelu 6595 . . . . . . . . 9  r  2nd ` 
<Q  r  *Q `  1st `
4543, 44syl6ibr 151 . . . . . . . 8  Q.  *Q `  <Q  r  1st `  r  2nd `
4645expcomd 1327 . . . . . . 7  Q.  1st `  *Q ` 
<Q  r  r  2nd `
4746imp 115 . . . . . 6  Q.  1st `  *Q `  <Q  r  r  2nd `
4847reximdv 2414 . . . . 5  Q.  1st `  r  Q.  *Q ` 
<Q  r  r  Q.  r  2nd `
4935, 48mpd 13 . . . 4  Q.  1st `  r  Q.  r  2nd `
5049rexlimiva 2422 . . 3  Q.  1st `  r 
Q.  r  2nd `
511, 26, 503syl 17 . 2  P.  r  Q.  r  2nd `
5225, 51jca 290 1  P.  q  Q.  q  1st `  r  Q.  r  2nd `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023  wrex 2301   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   1Qc1q 6265    +Q cplq 6266   *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-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:  recexprlempr  6604
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