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Theorem frecuzrdgsuc 9882
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9867 for the description of  G as the mapping from 
om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 28-May-2020.)
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 >. )
frecuzrdgtcl.3  |-  ( ph  ->  T  =  ran  R
)
Assertion
Ref Expression
frecuzrdgsuc  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
 B ) ) )
Distinct variable groups:    y, A    x, C, y    y, G    x, F, y    x, S, y    ph, x, y    x, B, y
Allowed substitution hints:    A( x)    R( x, y)    T( x, y)    G( x)

Proof of Theorem frecuzrdgsuc
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frec2uz.1 . . . . . . 7  |-  ( ph  ->  C  e.  ZZ )
21adantr 271 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
3 frec2uz.2 . . . . . 6  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
4 frecuzrdgrrn.a . . . . . . 7  |-  ( ph  ->  A  e.  S )
54adantr 271 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
6 frecuzrdgrrn.f . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
76adantlr 462 . . . . . 6  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
8 frecuzrdgrrn.2 . . . . . 6  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
9 peano2uz 9132 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
109adantl 272 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
112, 3, 5, 7, 8, 10frecuzrdglem 9879 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
12 frecuzrdgtcl.3 . . . . . 6  |-  ( ph  ->  T  =  ran  R
)
1312adantr 271 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  T  =  ran  R )
1411, 13eleqtrrd 2168 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T
)
151, 3, 4, 6, 8, 12frecuzrdgtcl 9880 . . . . . . 7  |-  ( ph  ->  T : ( ZZ>= `  C ) --> S )
16 ffun 5177 . . . . . . 7  |-  ( T : ( ZZ>= `  C
) --> S  ->  Fun  T )
1715, 16syl 14 . . . . . 6  |-  ( ph  ->  Fun  T )
18 funopfv 5357 . . . . . 6  |-  ( Fun 
T  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
1917, 18syl 14 . . . . 5  |-  ( ph  ->  ( <. ( B  + 
1 ) ,  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) )
>.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
2019adantr 271 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
2114, 20mpd 13 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) ) )
221, 3frec2uzf1od 9874 . . . . . . . . 9  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
23 f1ocnvdm 5574 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
2422, 23sylan 278 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  om )
252, 3, 24frec2uzsucd 9869 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
26 f1ocnvfv2 5571 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2722, 26sylan 278 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  =  B )
2827oveq1d 5681 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2925, 28eqtrd 2121 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
30 peano2 4423 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  suc  ( `' G `  B )  e.  om )
3124, 30syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  suc  ( `' G `  B )  e.  om )
32 f1ocnvfv 5572 . . . . . . 7  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  suc  ( `' G `  B )  e.  om )  ->  ( ( G `
 suc  ( `' G `  B )
)  =  ( B  +  1 )  -> 
( `' G `  ( B  +  1
) )  =  suc  ( `' G `  B ) ) )
3322, 31, 32syl2an2r 563 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  -> 
( `' G `  ( B  +  1
) )  =  suc  ( `' G `  B ) ) )
3429, 33mpd 13 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) )
3534fveq2d 5322 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3635fveq2d 5322 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3721, 36eqtrd 2121 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
38 1st2nd2 5959 . . . . . . . . . . 11  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
3938adantl 272 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4039fveq2d 5322 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
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 )
>. ) )
41 df-ov 5669 . . . . . . . . . . 11  |-  ( ( 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 )
>. )
42 xp1st 5950 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
4342adantl 272 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
44 xp2nd 5951 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
4544adantl 272 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
46 peano2uz 9132 . . . . . . . . . . . . . 14  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
4743, 46syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
48 oveq2 5674 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
4948eleq1d 2157 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
50 oveq1 5673 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
5150eleq1d 2157 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
5251ralbidv 2381 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  z
)  ->  ( A. y  e.  S  (
x F y )  e.  S  <->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
) )
536ralrimivva 2456 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
5453ad2antrr 473 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
5552, 54, 43rspcdva 2728 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
)
5649, 55, 45rspcdva 2728 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
57 opelxp 4481 . . . . . . . . . . . . 13  |-  ( <.
( ( 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
) )
5847, 56, 57sylanbrc 409 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
59 oveq1 5673 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
6059, 50opeq12d 3636 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
6148opeq2d 3635 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
62 eqid 2089 . . . . . . . . . . . . 13  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
6360, 61, 62ovmpt2g 5793 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) >. )
6443, 45, 58, 63syl3anc 1175 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  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 ) ) >. )
6541, 64syl5eqr 2135 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z ) F ( 2nd `  z
) ) >. )
6665, 58eqeltrd 2165 . . . . . . . . 9  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )  e.  ( ( ZZ>= `  C
)  X.  S ) )
6740, 66eqeltrd 2165 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S ) )
6867ralrimiva 2447 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A. z  e.  ( ( ZZ>= `  C
)  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  ( ( ZZ>= `  C
)  X.  S ) )
69 uzid 9094 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
702, 69syl 14 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ( ZZ>= `  C )
)
71 opelxp 4481 . . . . . . . 8  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
7270, 5, 71sylanbrc 409 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
73 frecsuc 6186 . . . . . . 7  |-  ( ( A. z  e.  ( ( ZZ>= `  C )  X.  S ) ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  ( ( ZZ>= `  C )  X.  S )  /\  <. C ,  A >.  e.  ( ( ZZ>= `  C )  X.  S )  /\  ( `' G `  B )  e.  om )  -> 
(frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) ) ) )
7468, 72, 24, 73syl3anc 1175 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) ) ) )
758fveq1i 5319 . . . . . 6  |-  ( R `
 suc  ( `' G `  B )
)  =  (frec ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )
768fveq1i 5319 . . . . . . 7  |-  ( R `
 ( `' G `  B ) )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) )
7776fveq2i 5321 . . . . . 6  |-  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  ( `' G `  B ) ) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  ( `' G `  B ) ) )
7874, 75, 773eqtr4g 2146 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) ) )
792, 3, 5, 7, 8, 24frec2uzrdg 9877 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. ( G `  ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
8079fveq2d 5322 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  ( `' G `  B ) ) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >. )
)
81 df-ov 5669 . . . . . 6  |-  ( ( G `  ( `' G `  B ) ) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >. )
8280, 81syl6eqr 2139 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  ( `' G `  B ) ) )  =  ( ( G `  ( `' G `  B ) ) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
832, 3, 24frec2uzuzd 9870 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  e.  ( ZZ>= `  C )
)
842, 3, 5, 7, 8frecuzrdgrrn 9876 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( `' G `  B )  e.  om )  ->  ( R `  ( `' G `  B )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
8524, 84mpdan 413 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
86 xp2nd 5951 . . . . . . 7  |-  ( ( R `  ( `' G `  B ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
8785, 86syl 14 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
8828, 10eqeltrd 2165 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  e.  ( ZZ>= `  C
) )
897caovclg 5811 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
9089, 83, 87caovcld 5812 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S )
91 opelxp 4481 . . . . . . 7  |-  ( <.
( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( ( ( G `
 ( `' G `  B ) )  +  1 )  e.  (
ZZ>= `  C )  /\  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S ) )
9288, 90, 91sylanbrc 409 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
93 oveq1 5673 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
94 oveq1 5673 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
9593, 94opeq12d 3636 . . . . . . 7  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
96 oveq2 5674 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
9796opeq2d 3635 . . . . . . 7  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
98 oveq1 5673 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
99 oveq1 5673 . . . . . . . . 9  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
10098, 99opeq12d 3636 . . . . . . . 8  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
101 oveq2 5674 . . . . . . . . 9  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
102101opeq2d 3635 . . . . . . . 8  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
103100, 102cbvmpt2v 5742 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
z  e.  ( ZZ>= `  C ) ,  w  e.  S  |->  <. (
z  +  1 ) ,  ( z F w ) >. )
10495, 97, 103ovmpt2g 5793 . . . . . 6  |-  ( ( ( G `  ( `' G `  B ) )  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S  /\  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( G `  ( `' G `  B )
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
10583, 87, 92, 104syl3anc 1175 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
10678, 82, 1053eqtrd 2125 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
107106fveq2d 5322 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( 2nd `  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
) )
108 op2ndg 5936 . . . 4  |-  ( ( ( ( G `  ( `' G `  B ) )  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S )  ->  ( 2nd `  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
10988, 90, 108syl2anc 404 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd ` 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
110107, 109eqtrd 2121 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
111 simpr 109 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  B  e.  ( ZZ>= `  C )
)
1122, 3, 5, 7, 8, 111frecuzrdglem 9879 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
113112, 13eleqtrrd 2168 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T
)
114 funopfv 5357 . . . . . . 7  |-  ( Fun 
T  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T  ->  ( T `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
11517, 114syl 14 . . . . . 6  |-  ( ph  ->  ( <. B ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>.  e.  T  ->  ( T `  B )  =  ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
116115adantr 271 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T  ->  ( T `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
117113, 116mpd 13 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
118117eqcomd 2094 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( T `  B ) )
11927, 118oveq12d 5684 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( T `  B ) ) )
12037, 110, 1193eqtrd 2125 1  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   A.wral 2360   <.cop 3453    |-> cmpt 3905   suc csuc 4201   omcom 4418    X. cxp 4450   `'ccnv 4451   ran crn 4453   Fun wfun 5022   -->wf 5024   -1-1-onto->wf1o 5027   ` cfv 5028  (class class class)co 5666    |-> cmpt2 5668   1stc1st 5923   2ndc2nd 5924  freccfrec 6169   1c1 7412    + caddc 7414   ZZcz 8811   ZZ>=cuz 9080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081
This theorem is referenced by:  iseqp1  9943
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