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Theorem frecuzrdgg 10129
Description: Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating  R at a natural number gives an ordered pair whose first element is the mapping of that natural number via  G. (Contributed by Jim Kingdon, 23-Apr-2022.)
Hypotheses
Ref Expression
frecuzrdgrclt.c  |-  ( ph  ->  C  e.  ZZ )
frecuzrdgrclt.a  |-  ( ph  ->  A  e.  S )
frecuzrdgrclt.t  |-  ( ph  ->  S  C_  T )
frecuzrdgrclt.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
frecuzrdgrclt.r  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
frecuzrdgg.n  |-  ( ph  ->  N  e.  om )
frecuzrdgg.g  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
Assertion
Ref Expression
frecuzrdgg  |-  ( ph  ->  ( 1st `  ( R `  N )
)  =  ( G `
 N ) )
Distinct variable groups:    x, C, y   
x, F, y    x, S, y    x, T, y    ph, x, y    x, G, y    x, R, y
Allowed substitution hints:    A( x, y)    N( x, y)

Proof of Theorem frecuzrdgg
Dummy variables  z  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgg.n . 2  |-  ( ph  ->  N  e.  om )
2 fveq2 5387 . . . . . 6  |-  ( w  =  (/)  ->  ( R `
 w )  =  ( R `  (/) ) )
32fveq2d 5391 . . . . 5  |-  ( w  =  (/)  ->  ( 1st `  ( R `  w
) )  =  ( 1st `  ( R `
 (/) ) ) )
4 fveq2 5387 . . . . 5  |-  ( w  =  (/)  ->  ( G `
 w )  =  ( G `  (/) ) )
53, 4eqeq12d 2130 . . . 4  |-  ( w  =  (/)  ->  ( ( 1st `  ( R `
 w ) )  =  ( G `  w )  <->  ( 1st `  ( R `  (/) ) )  =  ( G `  (/) ) ) )
65imbi2d 229 . . 3  |-  ( w  =  (/)  ->  ( (
ph  ->  ( 1st `  ( R `  w )
)  =  ( G `
 w ) )  <-> 
( ph  ->  ( 1st `  ( R `  (/) ) )  =  ( G `  (/) ) ) ) )
7 fveq2 5387 . . . . . 6  |-  ( w  =  k  ->  ( R `  w )  =  ( R `  k ) )
87fveq2d 5391 . . . . 5  |-  ( w  =  k  ->  ( 1st `  ( R `  w ) )  =  ( 1st `  ( R `  k )
) )
9 fveq2 5387 . . . . 5  |-  ( w  =  k  ->  ( G `  w )  =  ( G `  k ) )
108, 9eqeq12d 2130 . . . 4  |-  ( w  =  k  ->  (
( 1st `  ( R `  w )
)  =  ( G `
 w )  <->  ( 1st `  ( R `  k
) )  =  ( G `  k ) ) )
1110imbi2d 229 . . 3  |-  ( w  =  k  ->  (
( ph  ->  ( 1st `  ( R `  w
) )  =  ( G `  w ) )  <->  ( ph  ->  ( 1st `  ( R `
 k ) )  =  ( G `  k ) ) ) )
12 fveq2 5387 . . . . . 6  |-  ( w  =  suc  k  -> 
( R `  w
)  =  ( R `
 suc  k )
)
1312fveq2d 5391 . . . . 5  |-  ( w  =  suc  k  -> 
( 1st `  ( R `  w )
)  =  ( 1st `  ( R `  suc  k ) ) )
14 fveq2 5387 . . . . 5  |-  ( w  =  suc  k  -> 
( G `  w
)  =  ( G `
 suc  k )
)
1513, 14eqeq12d 2130 . . . 4  |-  ( w  =  suc  k  -> 
( ( 1st `  ( R `  w )
)  =  ( G `
 w )  <->  ( 1st `  ( R `  suc  k ) )  =  ( G `  suc  k ) ) )
1615imbi2d 229 . . 3  |-  ( w  =  suc  k  -> 
( ( ph  ->  ( 1st `  ( R `
 w ) )  =  ( G `  w ) )  <->  ( ph  ->  ( 1st `  ( R `  suc  k ) )  =  ( G `
 suc  k )
) ) )
17 fveq2 5387 . . . . . 6  |-  ( w  =  N  ->  ( R `  w )  =  ( R `  N ) )
1817fveq2d 5391 . . . . 5  |-  ( w  =  N  ->  ( 1st `  ( R `  w ) )  =  ( 1st `  ( R `  N )
) )
19 fveq2 5387 . . . . 5  |-  ( w  =  N  ->  ( G `  w )  =  ( G `  N ) )
2018, 19eqeq12d 2130 . . . 4  |-  ( w  =  N  ->  (
( 1st `  ( R `  w )
)  =  ( G `
 w )  <->  ( 1st `  ( R `  N
) )  =  ( G `  N ) ) )
2120imbi2d 229 . . 3  |-  ( w  =  N  ->  (
( ph  ->  ( 1st `  ( R `  w
) )  =  ( G `  w ) )  <->  ( ph  ->  ( 1st `  ( R `
 N ) )  =  ( G `  N ) ) ) )
22 frecuzrdgrclt.c . . . . 5  |-  ( ph  ->  C  e.  ZZ )
23 frecuzrdgrclt.a . . . . 5  |-  ( ph  ->  A  e.  S )
24 op1stg 6014 . . . . 5  |-  ( ( C  e.  ZZ  /\  A  e.  S )  ->  ( 1st `  <. C ,  A >. )  =  C )
2522, 23, 24syl2anc 406 . . . 4  |-  ( ph  ->  ( 1st `  <. C ,  A >. )  =  C )
26 frecuzrdgrclt.r . . . . . . 7  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
2726fveq1i 5388 . . . . . 6  |-  ( R `
 (/) )  =  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  (/) )
28 opexg 4118 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  A  e.  S )  -> 
<. C ,  A >.  e. 
_V )
29 frec0g 6260 . . . . . . . 8  |-  ( <. C ,  A >.  e. 
_V  ->  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. ) `  (/) )  = 
<. C ,  A >. )
3028, 29syl 14 . . . . . . 7  |-  ( ( C  e.  ZZ  /\  A  e.  S )  ->  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  (/) )  = 
<. C ,  A >. )
3122, 23, 30syl2anc 406 . . . . . 6  |-  ( ph  ->  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  (/) )  = 
<. C ,  A >. )
3227, 31syl5eq 2160 . . . . 5  |-  ( ph  ->  ( R `  (/) )  = 
<. C ,  A >. )
3332fveq2d 5391 . . . 4  |-  ( ph  ->  ( 1st `  ( R `  (/) ) )  =  ( 1st `  <. C ,  A >. )
)
34 frecuzrdgg.g . . . . 5  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
3522, 34frec2uz0d 10112 . . . 4  |-  ( ph  ->  ( G `  (/) )  =  C )
3625, 33, 353eqtr4d 2158 . . 3  |-  ( ph  ->  ( 1st `  ( R `  (/) ) )  =  ( G `  (/) ) )
3722, 34frec2uzf1od 10119 . . . . . . . . . . 11  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
38 f1of 5333 . . . . . . . . . . 11  |-  ( G : om -1-1-onto-> ( ZZ>= `  C )  ->  G : om --> ( ZZ>= `  C ) )
3937, 38syl 14 . . . . . . . . . 10  |-  ( ph  ->  G : om --> ( ZZ>= `  C ) )
4039ad2antlr 478 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  G : om --> ( ZZ>= `  C
) )
41 simpll 501 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  k  e.  om )
4240, 41ffvelrnd 5522 . . . . . . . 8  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( G `  k )  e.  ( ZZ>= `  C )
)
43 peano2uz 9327 . . . . . . . 8  |-  ( ( G `  k )  e.  ( ZZ>= `  C
)  ->  ( ( G `  k )  +  1 )  e.  ( ZZ>= `  C )
)
4442, 43syl 14 . . . . . . 7  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( G `  k
)  +  1 )  e.  ( ZZ>= `  C
) )
45 oveq2 5748 . . . . . . . . 9  |-  ( y  =  ( 2nd `  ( R `  k )
)  ->  ( ( G `  k ) F y )  =  ( ( G `  k ) F ( 2nd `  ( R `
 k ) ) ) )
4645eleq1d 2184 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( R `  k )
)  ->  ( (
( G `  k
) F y )  e.  S  <->  ( ( G `  k ) F ( 2nd `  ( R `  k )
) )  e.  S
) )
47 oveq1 5747 . . . . . . . . . . 11  |-  ( x  =  ( G `  k )  ->  (
x F y )  =  ( ( G `
 k ) F y ) )
4847eleq1d 2184 . . . . . . . . . 10  |-  ( x  =  ( G `  k )  ->  (
( x F y )  e.  S  <->  ( ( G `  k ) F y )  e.  S ) )
4948ralbidv 2412 . . . . . . . . 9  |-  ( x  =  ( G `  k )  ->  ( A. y  e.  S  ( x F y )  e.  S  <->  A. y  e.  S  ( ( G `  k ) F y )  e.  S ) )
50 frecuzrdgrclt.f . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
5150ralrimivva 2489 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
5251ad2antlr 478 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
5349, 52, 42rspcdva 2766 . . . . . . . 8  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  A. y  e.  S  ( ( G `  k ) F y )  e.  S )
54 frecuzrdgrclt.t . . . . . . . . . . . 12  |-  ( ph  ->  S  C_  T )
5522, 23, 54, 50, 26frecuzrdgrclt 10128 . . . . . . . . . . 11  |-  ( ph  ->  R : om --> ( (
ZZ>= `  C )  X.  S ) )
5655ad2antlr 478 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  R : om --> ( ( ZZ>= `  C )  X.  S
) )
5756, 41ffvelrnd 5522 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  k )  e.  ( ( ZZ>= `  C
)  X.  S ) )
58 xp2nd 6030 . . . . . . . . 9  |-  ( ( R `  k )  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  ( R `  k
) )  e.  S
)
5957, 58syl 14 . . . . . . . 8  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 2nd `  ( R `  k ) )  e.  S )
6046, 53, 59rspcdva 2766 . . . . . . 7  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( G `  k
) F ( 2nd `  ( R `  k
) ) )  e.  S )
61 op1stg 6014 . . . . . . 7  |-  ( ( ( ( G `  k )  +  1 )  e.  ( ZZ>= `  C )  /\  (
( G `  k
) F ( 2nd `  ( R `  k
) ) )  e.  S )  ->  ( 1st `  <. ( ( G `
 k )  +  1 ) ,  ( ( G `  k
) F ( 2nd `  ( R `  k
) ) ) >.
)  =  ( ( G `  k )  +  1 ) )
6244, 60, 61syl2anc 406 . . . . . 6  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 1st `  <. ( ( G `
 k )  +  1 ) ,  ( ( G `  k
) F ( 2nd `  ( R `  k
) ) ) >.
)  =  ( ( G `  k )  +  1 ) )
63 1st2nd2 6039 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
6463adantl 273 . . . . . . . . . . . . . . 15  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
6564fveq2d 5391 . . . . . . . . . . . . . 14  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
66 df-ov 5743 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  z ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
67 xp1st 6029 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
6867adantl 273 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( 1st `  z
)  e.  ( ZZ>= `  C ) )
6954ad3antlr 482 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  ->  S  C_  T )
70 xp2nd 6030 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
7170adantl 273 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( 2nd `  z
)  e.  S )
7269, 71sseldd 3066 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( 2nd `  z
)  e.  T )
73 peano2uz 9327 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
7468, 73syl 14 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( 1st `  z
)  +  1 )  e.  ( ZZ>= `  C
) )
75 oveq2 5748 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
7675eleq1d 2184 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
77 oveq1 5747 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
7877eleq1d 2184 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
7978ralbidv 2412 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  ( 1st `  z
)  ->  ( A. y  e.  S  (
x F y )  e.  S  <->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
) )
8051ad3antlr 482 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  ->  A. x  e.  ( ZZ>=
`  C ) A. y  e.  S  (
x F y )  e.  S )
8179, 80, 68rspcdva 2766 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  ->  A. y  e.  S  ( ( 1st `  z
) F y )  e.  S )
8276, 81, 71rspcdva 2766 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( 1st `  z
) F ( 2nd `  z ) )  e.  S )
83 opelxpi 4539 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  z
)  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
8474, 82, 83syl2anc 406 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  ->  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
85 oveq1 5747 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
8685, 77opeq12d 3681 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
8775opeq2d 3680 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
88 eqid 2115 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
8986, 87, 88ovmpog 5871 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1st `  z
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  z )  e.  T  /\  <. (
( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )  -> 
( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
9068, 72, 84, 89syl3anc 1199 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( 1st `  z
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  z
) )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
9166, 90syl5eqr 2162 . . . . . . . . . . . . . . 15  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z ) ) >. )
9291, 84eqeltrd 2192 . . . . . . . . . . . . . 14  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )  e.  (
( ZZ>= `  C )  X.  S ) )
9365, 92eqeltrd 2192 . . . . . . . . . . . . 13  |-  ( ( ( ( k  e. 
om  /\  ph )  /\  ( 1st `  ( R `
 k ) )  =  ( G `  k ) )  /\  z  e.  ( ( ZZ>=
`  C )  X.  S ) )  -> 
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  ( ( ZZ>= `  C
)  X.  S ) )
9493ralrimiva 2480 . . . . . . . . . . . 12  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  A. z  e.  ( ( ZZ>= `  C
)  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  ( ( ZZ>= `  C
)  X.  S ) )
95 uzid 9289 . . . . . . . . . . . . . . 15  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
9622, 95syl 14 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
97 opelxpi 4539 . . . . . . . . . . . . . 14  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
9896, 23, 97syl2anc 406 . . . . . . . . . . . . 13  |-  ( ph  -> 
<. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
9998ad2antlr 478 . . . . . . . . . . . 12  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
100 frecsuc 6270 . . . . . . . . . . . 12  |-  ( ( A. z  e.  ( ( ZZ>= `  C )  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S )  /\  <. C ,  A >.  e.  ( ( ZZ>= `  C )  X.  S )  /\  k  e.  om )  ->  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  k )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  k
) ) )
10194, 99, 41, 100syl3anc 1199 . . . . . . . . . . 11  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  k )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  k
) ) )
10226fveq1i 5388 . . . . . . . . . . 11  |-  ( R `
 suc  k )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  k )
10326fveq1i 5388 . . . . . . . . . . . 12  |-  ( R `
 k )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  k
)
104103fveq2i 5390 . . . . . . . . . . 11  |-  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  k
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  k
) )
105101, 102, 1043eqtr4g 2173 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  suc  k )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  ( R `
 k ) ) )
106 1st2nd2 6039 . . . . . . . . . . . 12  |-  ( ( R `  k )  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( R `  k )  =  <. ( 1st `  ( R `
 k ) ) ,  ( 2nd `  ( R `  k )
) >. )
10757, 106syl 14 . . . . . . . . . . 11  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  k )  =  <. ( 1st `  ( R `  k )
) ,  ( 2nd `  ( R `  k
) ) >. )
108107fveq2d 5391 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  k ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  ( R `  k
) ) ,  ( 2nd `  ( R `
 k ) )
>. ) )
109105, 108eqtrd 2148 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  suc  k )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( 1st `  ( R `  k ) ) ,  ( 2nd `  ( R `  k )
) >. ) )
110 simpr 109 . . . . . . . . . . 11  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 1st `  ( R `  k ) )  =  ( G `  k
) )
111110opeq1d 3679 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  <. ( 1st `  ( R `  k ) ) ,  ( 2nd `  ( R `  k )
) >.  =  <. ( G `  k ) ,  ( 2nd `  ( R `  k )
) >. )
112111fveq2d 5391 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( 1st `  ( R `  k
) ) ,  ( 2nd `  ( R `
 k ) )
>. )  =  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 k ) ,  ( 2nd `  ( R `  k )
) >. ) )
113109, 112eqtrd 2148 . . . . . . . 8  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  suc  k )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( G `  k ) ,  ( 2nd `  ( R `  k )
) >. ) )
114 df-ov 5743 . . . . . . . . 9  |-  ( ( G `  k ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  k )
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 k ) ,  ( 2nd `  ( R `  k )
) >. )
11554ad2antlr 478 . . . . . . . . . . 11  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  S  C_  T )
116115, 59sseldd 3066 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 2nd `  ( R `  k ) )  e.  T )
117 opelxpi 4539 . . . . . . . . . . 11  |-  ( ( ( ( G `  k )  +  1 )  e.  ( ZZ>= `  C )  /\  (
( G `  k
) F ( 2nd `  ( R `  k
) ) )  e.  S )  ->  <. (
( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >.  e.  ( ( ZZ>= `  C )  X.  S ) )
11844, 60, 117syl2anc 406 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  <. (
( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >.  e.  ( ( ZZ>= `  C )  X.  S ) )
119 oveq1 5747 . . . . . . . . . . . 12  |-  ( x  =  ( G `  k )  ->  (
x  +  1 )  =  ( ( G `
 k )  +  1 ) )
120119, 47opeq12d 3681 . . . . . . . . . . 11  |-  ( x  =  ( G `  k )  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F y ) >. )
12145opeq2d 3680 . . . . . . . . . . 11  |-  ( y  =  ( 2nd `  ( R `  k )
)  ->  <. ( ( G `  k )  +  1 ) ,  ( ( G `  k ) F y ) >.  =  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >. )
122120, 121, 88ovmpog 5871 . . . . . . . . . 10  |-  ( ( ( G `  k
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  k ) )  e.  T  /\  <. (
( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >.  e.  ( ( ZZ>= `  C )  X.  S ) )  -> 
( ( G `  k ) ( x  e.  ( ZZ>= `  C
) ,  y  e.  T  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  k )
) )  =  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >. )
12342, 116, 118, 122syl3anc 1199 . . . . . . . . 9  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( G `  k
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  k )
) )  =  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >. )
124114, 123syl5eqr 2162 . . . . . . . 8  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  (
( x  e.  (
ZZ>= `  C ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 k ) ,  ( 2nd `  ( R `  k )
) >. )  =  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >. )
125113, 124eqtrd 2148 . . . . . . 7  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( R `  suc  k )  =  <. ( ( G `
 k )  +  1 ) ,  ( ( G `  k
) F ( 2nd `  ( R `  k
) ) ) >.
)
126125fveq2d 5391 . . . . . 6  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 1st `  ( R `  suc  k ) )  =  ( 1st `  <. ( ( G `  k
)  +  1 ) ,  ( ( G `
 k ) F ( 2nd `  ( R `  k )
) ) >. )
)
12722ad2antlr 478 . . . . . . 7  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  C  e.  ZZ )
128127, 34, 41frec2uzsucd 10114 . . . . . 6  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( G `  suc  k )  =  ( ( G `
 k )  +  1 ) )
12962, 126, 1283eqtr4d 2158 . . . . 5  |-  ( ( ( k  e.  om  /\ 
ph )  /\  ( 1st `  ( R `  k ) )  =  ( G `  k
) )  ->  ( 1st `  ( R `  suc  k ) )  =  ( G `  suc  k ) )
130129exp31 359 . . . 4  |-  ( k  e.  om  ->  ( ph  ->  ( ( 1st `  ( R `  k
) )  =  ( G `  k )  ->  ( 1st `  ( R `  suc  k ) )  =  ( G `
 suc  k )
) ) )
131130a2d 26 . . 3  |-  ( k  e.  om  ->  (
( ph  ->  ( 1st `  ( R `  k
) )  =  ( G `  k ) )  ->  ( ph  ->  ( 1st `  ( R `  suc  k ) )  =  ( G `
 suc  k )
) ) )
1326, 11, 16, 21, 36, 131finds 4482 . 2  |-  ( N  e.  om  ->  ( ph  ->  ( 1st `  ( R `  N )
)  =  ( G `
 N ) ) )
1331, 132mpcom 36 1  |-  ( ph  ->  ( 1st `  ( R `  N )
)  =  ( G `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658    C_ wss 3039   (/)c0 3331   <.cop 3498    |-> cmpt 3957   suc csuc 4255   omcom 4472    X. cxp 4505   -->wf 5087   -1-1-onto->wf1o 5090   ` cfv 5091  (class class class)co 5740    e. cmpo 5742   1stc1st 6002   2ndc2nd 6003  freccfrec 6253   1c1 7585    + caddc 7587   ZZcz 9005   ZZ>=cuz 9275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8678  df-n0 8929  df-z 9006  df-uz 9276
This theorem is referenced by:  frecuzrdgdomlem  10130  frecuzrdgfunlem  10132  frecuzrdgsuctlem  10136
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