ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frecuzrdgrrn Structured version   Unicode version

Theorem frecuzrdgrrn 8855
Description: The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 
S. (Contributed by Jim Kingdon, 27-May-2020.)
Hypotheses
Ref Expression
frec2uz.1  C  ZZ
frec2uz.2  G frec  ZZ  |->  +  1 ,  C
uzrdg.s  S  V
uzrdg.a  S
uzrdg.f  ZZ>= `  C  S  F  S
uzrdg.2  R frec 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.
Assertion
Ref Expression
frecuzrdgrrn  D  om  R `  D  ZZ>= `  C  X.  S
Distinct variable groups:   ,   , C,   , G   , F,   , S,   ,,
Allowed substitution hints:   ()    D(,)    R(,)    G()    V(,)

Proof of Theorem frecuzrdgrrn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 uzrdg.2 . . 3  R frec 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >.
21fveq1i 5122 . 2  R `
 D frec  ZZ>= `  C ,  S  |->  <.  + 
1 ,  F
>. ,  <. C ,  >. `  D
3 zex 8010 . . . . . . . 8  ZZ  _V
4 uzssz 8248 . . . . . . . 8  ZZ>= `  C  C_  ZZ
53, 4ssexi 3886 . . . . . . 7  ZZ>= `  C  _V
65a1i 9 . . . . . 6  D  om  ZZ>= `  C  _V
7 uzrdg.s . . . . . . 7  S  V
87adantr 261 . . . . . 6  D  om  S  V
9 mpt2exga 5777 . . . . . 6  ZZ>= `  C  _V  S  V  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >.  _V
106, 8, 9syl2anc 391 . . . . 5  D  om  ZZ>= `  C ,  S  |-> 
<.  +  1 ,  F >.  _V
11 vex 2554 . . . . . 6 
_V
1211a1i 9 . . . . 5  D  om  _V
13 fvexg 5137 . . . . 5 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >.  _V  _V  ZZ>=
`  C ,  S  |->  <.  +  1 ,  F >. `

_V
1410, 12, 13syl2anc 391 . . . 4  D  om  ZZ>= `  C ,  S  |->  <.  + 
1 ,  F
>. `  _V
1514alrimiv 1751 . . 3  D  om 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. `

_V
16 frec2uz.1 . . . . . 6  C  ZZ
17 uzid 8243 . . . . . 6  C  ZZ  C  ZZ>= `  C
1816, 17syl 14 . . . . 5  C  ZZ>= `  C
19 uzrdg.a . . . . 5  S
20 opelxp 4317 . . . . 5  <. C ,  >.  ZZ>= `  C  X.  S  C  ZZ>= `  C  S
2118, 19, 20sylanbrc 394 . . . 4  <. C ,  >.  ZZ>= `  C  X.  S
2221adantr 261 . . 3  D  om  <. C ,  >.  ZZ>= `  C  X.  S
23 1st2nd2 5743 . . . . . . 7  ZZ>= `  C  X.  S  <. 1st `  ,  2nd `  >.
24 fveq2 5121 . . . . . . . 8  <. 1st `  ,  2nd `  >.  ZZ>=
`  C ,  S  |->  <.  +  1 ,  F >. `
 ZZ>= `  C ,  S  |-> 
<.  +  1 ,  F >. `
 <. 1st `  ,  2nd `  >.
25 df-ov 5458 . . . . . . . 8  1st ` 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. 2nd `  ZZ>=
`  C ,  S  |->  <.  +  1 ,  F >. `
 <. 1st `  ,  2nd `  >.
2624, 25syl6eqr 2087 . . . . . . 7  <. 1st `  ,  2nd `  >.  ZZ>=
`  C ,  S  |->  <.  +  1 ,  F >. `
 1st ` 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. 2nd `
2723, 26syl 14 . . . . . 6  ZZ>= `  C  X.  S  ZZ>= `  C ,  S  |->  <.  + 
1 ,  F
>. `  1st `  ZZ>= `  C ,  S  |->  <.  + 
1 ,  F
>. 2nd `
2827adantl 262 . . . . 5  D  om  ZZ>= `  C  X.  S  ZZ>= `  C ,  S  |-> 
<.  +  1 ,  F >. `
 1st ` 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. 2nd `
29 xp1st 5734 . . . . . . 7  ZZ>= `  C  X.  S  1st `  ZZ>= `  C
3029adantl 262 . . . . . 6  D  om  ZZ>= `  C  X.  S  1st `  ZZ>= `  C
31 xp2nd 5735 . . . . . . 7  ZZ>= `  C  X.  S  2nd `  S
3231adantl 262 . . . . . 6  D  om  ZZ>= `  C  X.  S  2nd `  S
33 peano2uz 8282 . . . . . . . 8  1st `  ZZ>= `  C  1st `  +  1  ZZ>= `  C
3430, 33syl 14 . . . . . . 7  D  om  ZZ>= `  C  X.  S  1st `  +  1  ZZ>= `  C
35 uzrdg.f . . . . . . . . . 10  ZZ>= `  C  S  F  S
3635ralrimivva 2395 . . . . . . . . 9 
ZZ>= `  C  S  F  S
3736ad2antrr 457 . . . . . . . 8  D  om  ZZ>= `  C  X.  S 
ZZ>= `  C  S  F  S
38 oveq1 5462 . . . . . . . . . . 11  1st `  F  1st `  F
3938eleq1d 2103 . . . . . . . . . 10  1st `  F  S  1st `  F  S
40 oveq2 5463 . . . . . . . . . . 11  2nd `  1st `  F  1st `  F 2nd `
4140eleq1d 2103 . . . . . . . . . 10  2nd `  1st `  F  S  1st `  F 2nd `  S
4239, 41rspc2v 2656 . . . . . . . . 9  1st `  ZZ>= `  C  2nd `  S  ZZ>=
`  C  S  F  S  1st `  F 2nd `  S
4330, 32, 42syl2anc 391 . . . . . . . 8  D  om  ZZ>= `  C  X.  S  ZZ>= `  C  S  F  S  1st `  F 2nd `  S
4437, 43mpd 13 . . . . . . 7  D  om  ZZ>= `  C  X.  S  1st `  F 2nd `  S
45 opelxp 4317 . . . . . . 7  <. 1st `  +  1 ,  1st `  F 2nd `  >.  ZZ>= `  C  X.  S  1st `  +  1  ZZ>= `  C  1st `  F 2nd `  S
4634, 44, 45sylanbrc 394 . . . . . 6  D  om  ZZ>= `  C  X.  S  <. 1st `  +  1 ,  1st `  F 2nd `  >.  ZZ>= `  C  X.  S
47 oveq1 5462 . . . . . . . 8  1st `  + 
1  1st `  +  1
4847, 38opeq12d 3548 . . . . . . 7  1st `  <.  + 
1 ,  F
>.  <. 1st `  +  1 ,  1st `  F >.
4940opeq2d 3547 . . . . . . 7  2nd `  <. 1st `  +  1 ,  1st `  F >. 
<. 1st `  +  1 ,  1st `  F 2nd `  >.
50 eqid 2037 . . . . . . 7  ZZ>= `  C ,  S  |->  <.  + 
1 ,  F
>.  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >.
5148, 49, 50ovmpt2g 5577 . . . . . 6  1st `  ZZ>= `  C  2nd `  S  <. 1st `  +  1 ,  1st `  F 2nd `  >.  ZZ>= `  C  X.  S  1st ` 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. 2nd `  <. 1st `  +  1 ,  1st `  F 2nd `  >.
5230, 32, 46, 51syl3anc 1134 . . . . 5  D  om  ZZ>= `  C  X.  S  1st ` 
ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. 2nd `  <. 1st `  +  1 ,  1st `  F 2nd `  >.
5328, 52eqtrd 2069 . . . 4  D  om  ZZ>= `  C  X.  S  ZZ>= `  C ,  S  |-> 
<.  +  1 ,  F >. `

<. 1st `  +  1 ,  1st `  F 2nd `  >.
5453, 46eqeltrd 2111 . . 3  D  om  ZZ>= `  C  X.  S  ZZ>= `  C ,  S  |-> 
<.  +  1 ,  F >. `
 ZZ>= `  C  X.  S
55 simpr 103 . . 3  D  om  D  om
5615, 22, 54, 55freccl 5932 . 2  D  om frec  ZZ>= `  C ,  S  |->  <.  +  1 ,  F >. ,  <. C ,  >. `  D  ZZ>= `  C  X.  S
572, 56syl5eqel 2121 1  D  om  R `  D  ZZ>= `  C  X.  S
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551   <.cop 3370    |-> cmpt 3809   omcom 4256    X. cxp 4286   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708  freccfrec 5917   1c1 6692    + caddc 6694   ZZcz 8001   ZZ>=cuz 8229
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-bnd 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  ax-cnex 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-addcom 6763  ax-addass 6765  ax-distr 6767  ax-i2m1 6768  ax-0id 6771  ax-rnegex 6772  ax-cnre 6774  ax-pre-ltirr 6775  ax-pre-ltwlin 6776  ax-pre-lttrn 6777  ax-pre-ltadd 6779
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-nel 2204  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-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  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-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-1r 6640  df-0 6698  df-1 6699  df-r 6701  df-lt 6704  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842  df-le 6843  df-sub 6961  df-neg 6962  df-inn 7676  df-n0 7938  df-z 8002  df-uz 8230
This theorem is referenced by:  frec2uzrdg  8856  frecuzrdgfn  8859  frecuzrdgcl  8860  frecuzrdgsuc  8862
  Copyright terms: Public domain W3C validator