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Theorem frecuzrdgrrn 9780
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, 28-Mar-2022.)
Hypotheses
Ref Expression
frec2uz.1  |-  ( ph  ->  C  e.  ZZ )
frec2uz.2  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
frecuzrdgrrn.a  |-  ( ph  ->  A  e.  S )
frecuzrdgrrn.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
frecuzrdgrrn.2  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
Assertion
Ref Expression
frecuzrdgrrn  |-  ( (
ph  /\  D  e.  om )  ->  ( R `  D )  e.  ( ( ZZ>= `  C )  X.  S ) )
Distinct variable groups:    y, A    x, C, y    y, G    x, F, y    x, S, y    ph, x, y
Allowed substitution hints:    A( x)    D( x, y)    R( x, y)    G( x)

Proof of Theorem frecuzrdgrrn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frecuzrdgrrn.2 . . 3  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
21fveq1i 5290 . 2  |-  ( R `
 D )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  D
)
3 frec2uz.1 . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
4 uzid 9002 . . . . . 6  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
53, 4syl 14 . . . . 5  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
6 frecuzrdgrrn.a . . . . 5  |-  ( ph  ->  A  e.  S )
7 opelxp 4457 . . . . 5  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
85, 6, 7sylanbrc 408 . . . 4  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
98adantr 270 . . 3  |-  ( (
ph  /\  D  e.  om )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
10 1st2nd2 5927 . . . . . . 7  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
11 fveq2 5289 . . . . . . . 8  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  z )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
12 df-ov 5637 . . . . . . . 8  |-  ( ( 1st `  z ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1311, 12syl6eqr 2138 . . . . . . 7  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  z )  =  ( ( 1st `  z ) ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  z
) ) )
1410, 13syl 14 . . . . . 6  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  ( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) ) )
1514adantl 271 . . . . 5  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  ( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) ) )
16 xp1st 5918 . . . . . . 7  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
1716adantl 271 . . . . . 6  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
18 xp2nd 5919 . . . . . . 7  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
1918adantl 271 . . . . . 6  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
20 peano2uz 9040 . . . . . . . 8  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
2117, 20syl 14 . . . . . . 7  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
22 frecuzrdgrrn.f . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
2322ralrimivva 2455 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
2423ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
25 oveq1 5641 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
2625eleq1d 2156 . . . . . . . . . 10  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
27 oveq2 5642 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
2827eleq1d 2156 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
2926, 28rspc2v 2733 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  S )  ->  ( A. x  e.  ( ZZ>=
`  C ) A. y  e.  S  (
x F y )  e.  S  ->  (
( 1st `  z
) F ( 2nd `  z ) )  e.  S ) )
3017, 19, 29syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
3124, 30mpd 13 . . . . . . 7  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
32 opelxp 4457 . . . . . . 7  |-  ( <.
( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S )  <->  ( (
( 1st `  z
)  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
3321, 31, 32sylanbrc 408 . . . . . 6  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
34 oveq1 5641 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
3534, 25opeq12d 3625 . . . . . . 7  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
3627opeq2d 3624 . . . . . . 7  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
37 eqid 2088 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
3835, 36, 37ovmpt2g 5761 . . . . . 6  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  S  /\  <. (
( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )  -> 
( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
3917, 19, 33, 38syl3anc 1174 . . . . 5  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
4015, 39eqtrd 2120 . . . 4  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
4140, 33eqeltrd 2164 . . 3  |-  ( ( ( ph  /\  D  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S ) )
42 simpr 108 . . 3  |-  ( (
ph  /\  D  e.  om )  ->  D  e.  om )
439, 41, 42freccl 6150 . 2  |-  ( (
ph  /\  D  e.  om )  ->  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  D
)  e.  ( (
ZZ>= `  C )  X.  S ) )
442, 43syl5eqel 2174 1  |-  ( (
ph  /\  D  e.  om )  ->  ( R `  D )  e.  ( ( ZZ>= `  C )  X.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   <.cop 3444    |-> cmpt 3891   omcom 4395    X. cxp 4426   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892  freccfrec 6137   1c1 7330    + caddc 7332   ZZcz 8720   ZZ>=cuz 8988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989
This theorem is referenced by:  frec2uzrdg  9781  frecuzrdgtcl  9784  frecuzrdgsuc  9786
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