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Theorem recexprlemloc 6603
Description: is located. 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
recexprlemloc  P.  q  Q.  r  Q.  q  <Q 
r  q  1st `  r  2nd `
Distinct variable groups:    r, q,,,   , q,
r,,

Proof of Theorem recexprlemloc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6458 . . . . . . . . 9  P.  <. 1st `  ,  2nd `  >.  P.
2 prnmaxl 6471 . . . . . . . . 9 
<. 1st `  ,  2nd `  >.  P.  *Q `  r  1st `  1st `  *Q `  r 
<Q
31, 2sylan 267 . . . . . . . 8  P.  *Q `  r  1st `  1st `  *Q `  r  <Q
43adantlr 446 . . . . . . 7  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r 
<Q
5 simprr 484 . . . . . . . . . 10  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  r  <Q
6 elprnql 6464 . . . . . . . . . . . . . 14 
<. 1st `  ,  2nd `  >.  P.  1st `  Q.
71, 6sylan 267 . . . . . . . . . . . . 13  P.  1st `  Q.
87ad2ant2r 478 . . . . . . . . . . . 12  P.  q  <Q  r  1st `  *Q `  r  <Q  Q.
98adantlr 446 . . . . . . . . . . 11  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  Q.
10 recrecnq 6378 . . . . . . . . . . 11  Q.  *Q `  *Q `
119, 10syl 14 . . . . . . . . . 10  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  *Q `
125, 11breqtrrd 3781 . . . . . . . . 9  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  r  <Q  *Q `  *Q `
13 recclnq 6376 . . . . . . . . . . 11  Q.  *Q ` 
Q.
149, 13syl 14 . . . . . . . . . 10  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  Q.
15 ltrelnq 6349 . . . . . . . . . . . . . 14  <Q  C_  Q.  X.  Q.
1615brel 4335 . . . . . . . . . . . . 13  q 
<Q  r 
q  Q.  r  Q.
1716adantl 262 . . . . . . . . . . . 12  P.  q  <Q  r  q  Q.  r  Q.
1817ad2antrr 457 . . . . . . . . . . 11  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  q  Q.  r  Q.
1918simprd 107 . . . . . . . . . 10  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  r  Q.
20 ltrnqg 6403 . . . . . . . . . 10  *Q `  Q.  r  Q.  *Q ` 
<Q  r  *Q `  r  <Q  *Q `  *Q `
2114, 19, 20syl2anc 391 . . . . . . . . 9  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  <Q  r  *Q `  r  <Q  *Q `  *Q `
2212, 21mpbird 156 . . . . . . . 8  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  <Q  r
23 simprl 483 . . . . . . . . 9  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  1st `
2411, 23eqeltrd 2111 . . . . . . . 8  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  *Q `  1st `
25 breq1 3758 . . . . . . . . . . . 12  *Q `  <Q  r  *Q `  <Q  r
26 fveq2 5121 . . . . . . . . . . . . 13  *Q `  *Q `  *Q `  *Q `
2726eleq1d 2103 . . . . . . . . . . . 12  *Q `  *Q `  1st `  *Q `  *Q `  1st `
2825, 27anbi12d 442 . . . . . . . . . . 11  *Q `  <Q  r  *Q `  1st `  *Q `  <Q  r  *Q `  *Q `  1st `
2928spcegv 2635 . . . . . . . . . 10  *Q `  Q.  *Q `  <Q  r  *Q `  *Q `  1st `  <Q  r  *Q `  1st `
30 recexpr.1 . . . . . . . . . . 11  <. {  |  <Q  *Q `  2nd `  } ,  {  |  <Q  *Q `  1st `  } >.
3130recexprlemelu 6595 . . . . . . . . . 10  r  2nd ` 
<Q  r  *Q `  1st `
3229, 31syl6ibr 151 . . . . . . . . 9  *Q `  Q.  *Q `  <Q  r  *Q `  *Q `  1st `  r  2nd `
3314, 32syl 14 . . . . . . . 8  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  *Q `  <Q  r  *Q `  *Q `  1st `  r  2nd `
3422, 24, 33mp2and 409 . . . . . . 7  P.  q  <Q  r  *Q `  r  1st `  1st `  *Q `  r  <Q  r  2nd `
354, 34rexlimddv 2431 . . . . . 6  P.  q  <Q  r  *Q `  r  1st `  r  2nd `
3635olcd 652 . . . . 5  P.  q  <Q  r  *Q `  r  1st `  q  1st `  r  2nd `
37 prnminu 6472 . . . . . . . . 9 
<. 1st `  ,  2nd `  >.  P.  *Q `  q  2nd `  2nd ` 
<Q  *Q `  q
381, 37sylan 267 . . . . . . . 8  P.  *Q `  q  2nd `  2nd `  <Q  *Q `  q
3938adantlr 446 . . . . . . 7  P.  q  <Q  r  *Q `  q  2nd `  2nd ` 
<Q  *Q `  q
40 elprnqu 6465 . . . . . . . . . . . . . 14 
<. 1st `  ,  2nd `  >.  P.  2nd `  Q.
411, 40sylan 267 . . . . . . . . . . . . 13  P.  2nd `  Q.
4241adantlr 446 . . . . . . . . . . . 12  P.  q  <Q  r  2nd `  Q.
4342ad2ant2r 478 . . . . . . . . . . 11  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  Q.
44 recrecnq 6378 . . . . . . . . . . 11  Q.  *Q `  *Q `
4543, 44syl 14 . . . . . . . . . 10  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  *Q `  *Q `
46 simprr 484 . . . . . . . . . 10  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  <Q  *Q `  q
4745, 46eqbrtrd 3775 . . . . . . . . 9  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  *Q `  *Q ` 
<Q  *Q `  q
4817ad2antrr 457 . . . . . . . . . . 11  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  Q.  r  Q.
4948simpld 105 . . . . . . . . . 10  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  Q.
50 recclnq 6376 . . . . . . . . . . 11  Q.  *Q ` 
Q.
5143, 50syl 14 . . . . . . . . . 10  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  *Q `  Q.
52 ltrnqg 6403 . . . . . . . . . 10  q  Q.  *Q `  Q.  q  <Q  *Q `  *Q `  *Q `  <Q  *Q `  q
5349, 51, 52syl2anc 391 . . . . . . . . 9  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  <Q  *Q `  *Q `  *Q `  <Q  *Q `  q
5447, 53mpbird 156 . . . . . . . 8  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  <Q  *Q `
55 simprl 483 . . . . . . . . 9  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  2nd `
5645, 55eqeltrd 2111 . . . . . . . 8  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  *Q `  *Q `  2nd `
57 breq2 3759 . . . . . . . . . . . 12  *Q ` 
q  <Q  q  <Q  *Q `
58 fveq2 5121 . . . . . . . . . . . . 13  *Q `  *Q `  *Q `  *Q `
5958eleq1d 2103 . . . . . . . . . . . 12  *Q `  *Q `  2nd `  *Q `  *Q `  2nd `
6057, 59anbi12d 442 . . . . . . . . . . 11  *Q `  q  <Q  *Q `  2nd `  q  <Q  *Q `  *Q `  *Q `  2nd `
6160spcegv 2635 . . . . . . . . . 10  *Q `  Q.  q  <Q  *Q `  *Q `  *Q `  2nd `  q  <Q  *Q `  2nd `
6230recexprlemell 6594 . . . . . . . . . 10  q  1st `  q 
<Q  *Q `  2nd `
6361, 62syl6ibr 151 . . . . . . . . 9  *Q `  Q.  q  <Q  *Q `  *Q `  *Q `  2nd `  q  1st `
6451, 63syl 14 . . . . . . . 8  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  <Q  *Q `  *Q `  *Q `  2nd `  q  1st `
6554, 56, 64mp2and 409 . . . . . . 7  P.  q  <Q  r  *Q `  q  2nd `  2nd `  <Q  *Q `  q  q  1st `
6639, 65rexlimddv 2431 . . . . . 6  P.  q  <Q  r  *Q `  q  2nd `  q  1st `
6766orcd 651 . . . . 5  P.  q  <Q  r  *Q `  q  2nd `  q  1st `  r  2nd `
68 ltrnqi 6404 . . . . . 6  q 
<Q  r  *Q `  r  <Q  *Q `  q
69 prloc 6474 . . . . . 6 
<. 1st `  ,  2nd `  >.  P.  *Q `  r  <Q  *Q `  q  *Q `  r  1st `  *Q `  q  2nd `
701, 68, 69syl2an 273 . . . . 5  P.  q  <Q  r  *Q `  r  1st `  *Q `  q  2nd `
7136, 67, 70mpjaodan 710 . . . 4  P.  q  <Q  r  q  1st `  r  2nd `
7271ex 108 . . 3  P. 
q  <Q  r 
q  1st `  r  2nd `
7372ralrimivw 2387 . 2  P.  r  Q. 
q  <Q  r 
q  1st `  r  2nd `
7473ralrimivw 2387 1  P.  q  Q.  r  Q.  q  <Q 
r  q  1st `  r  2nd `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242  wex 1378   wcel 1390   {cab 2023  wral 2300  wrex 2301   <.cop 3370   class class class wbr 3755   ` 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-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-mi 6290  df-lti 6291  df-mpq 6329  df-enq 6331  df-nqqs 6332  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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