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Theorem frecuzrdgsuc 10142
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10127 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 274 . . . . . 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 274 . . . . . 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 468 . . . . . 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 9334 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
109adantl 275 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
112, 3, 5, 7, 8, 10frecuzrdglem 10139 . . . . 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 274 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  T  =  ran  R )
1411, 13eleqtrrd 2197 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T
)
151, 3, 4, 6, 8, 12frecuzrdgtcl 10140 . . . . . . 7  |-  ( ph  ->  T : ( ZZ>= `  C ) --> S )
16 ffun 5245 . . . . . . 7  |-  ( T : ( ZZ>= `  C
) --> S  ->  Fun  T )
1715, 16syl 14 . . . . . 6  |-  ( ph  ->  Fun  T )
18 funopfv 5429 . . . . . 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 274 . . . 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 10134 . . . . . . . . 9  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
23 f1ocnvdm 5650 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
2422, 23sylan 281 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  om )
252, 3, 24frec2uzsucd 10129 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
26 f1ocnvfv2 5647 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2722, 26sylan 281 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  =  B )
2827oveq1d 5757 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2925, 28eqtrd 2150 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
30 peano2 4479 . . . . . . . 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 5648 . . . . . . 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 569 . . . . . 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 5393 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3635fveq2d 5393 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3721, 36eqtrd 2150 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
38 1st2nd2 6041 . . . . . . . . . . 11  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
3938adantl 275 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4039fveq2d 5393 . . . . . . . . 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 5745 . . . . . . . . . . 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 6031 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
4342adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
44 xp2nd 6032 . . . . . . . . . . . . 13  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
4544adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
46 peano2uz 9334 . . . . . . . . . . . . . 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 5750 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
4948eleq1d 2186 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
50 oveq1 5749 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
5150eleq1d 2186 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
5251ralbidv 2414 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  z
)  ->  ( A. y  e.  S  (
x F y )  e.  S  <->  A. y  e.  S  ( ( 1st `  z ) F y )  e.  S
) )
536ralrimivva 2491 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
5453ad2antrr 479 . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
57 opelxp 4539 . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 5749 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
6059, 50opeq12d 3683 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
6148opeq2d 3682 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
62 eqid 2117 . . . . . . . . . . . . 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, 62ovmpog 5873 . . . . . . . . . . . 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 1201 . . . . . . . . . . 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 2164 . . . . . . . . . 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 2194 . . . . . . . . 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 2194 . . . . . . . 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 2482 . . . . . . 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 9296 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
702, 69syl 14 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ( ZZ>= `  C )
)
71 opelxp 4539 . . . . . . . 8  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
7270, 5, 71sylanbrc 413 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
73 frecsuc 6272 . . . . . . 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 1201 . . . . . 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 5390 . . . . . 6  |-  ( R `
 suc  ( `' G `  B )
)  =  (frec ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )
768fveq1i 5390 . . . . . . 7  |-  ( R `
 ( `' G `  B ) )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) )
7776fveq2i 5392 . . . . . 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 2175 . . . . 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 10137 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. ( G `  ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
8079fveq2d 5393 . . . . . 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 5745 . . . . . 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 2168 . . . . 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 10130 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  e.  ( ZZ>= `  C )
)
842, 3, 5, 7, 8frecuzrdgrrn 10136 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( `' G `  B )  e.  om )  ->  ( R `  ( `' G `  B )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
8524, 84mpdan 417 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
86 xp2nd 6032 . . . . . . 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 2194 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  e.  ( ZZ>= `  C
) )
897caovclg 5891 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
9089, 83, 87caovcld 5892 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S )
91 opelxp 4539 . . . . . . 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 413 . . . . . 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 5749 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
94 oveq1 5749 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
9593, 94opeq12d 3683 . . . . . . 7  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
96 oveq2 5750 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
9796opeq2d 3682 . . . . . . 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 5749 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
99 oveq1 5749 . . . . . . . . 9  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
10098, 99opeq12d 3683 . . . . . . . 8  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
101 oveq2 5750 . . . . . . . . 9  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
102101opeq2d 3682 . . . . . . . 8  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
103100, 102cbvmpov 5819 . . . . . . 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, 103ovmpog 5873 . . . . . 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 1201 . . . . 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 2154 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
107106fveq2d 5393 . . 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 6017 . . . 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 408 . . 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 2150 . 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 10139 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
113112, 13eleqtrrd 2197 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T
)
114 funopfv 5429 . . . . . . 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 274 . . . . 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 2123 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( T `  B ) )
11927, 118oveq12d 5760 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( T `  B ) ) )
12037, 110, 1193eqtrd 2154 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 1316    e. wcel 1465   A.wral 2393   <.cop 3500    |-> cmpt 3959   suc csuc 4257   omcom 4474    X. cxp 4507   `'ccnv 4508   ran crn 4510   Fun wfun 5087   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005  freccfrec 6255   1c1 7589    + caddc 7591   ZZcz 9012   ZZ>=cuz 9282
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283
This theorem is referenced by: (None)
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