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Theorem frec2uzrdg 9543
Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function 
F ( x ,  y ) and initial value  A. This lemma shows that evaluating  R at an element of  om gives an ordered pair whose first element is the index (translated from  om to  ( ZZ>= `  C )). See comment in frec2uz0d 9533 which describes  G and the index translation. (Contributed by Jim Kingdon, 24-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 >. )
frec2uzrdg.b  |-  ( ph  ->  B  e.  om )
Assertion
Ref Expression
frec2uzrdg  |-  ( ph  ->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
Distinct variable groups:    y, A    x, C, y    y, G    x, F, y    x, S, y    ph, x, y
Allowed substitution hints:    A( x)    B( x, y)    R( x, y)    G( x)

Proof of Theorem frec2uzrdg
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frec2uzrdg.b . 2  |-  ( ph  ->  B  e.  om )
2 fveq2 5229 . . . . 5  |-  ( z  =  B  ->  ( R `  z )  =  ( R `  B ) )
3 fveq2 5229 . . . . . 6  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
42fveq2d 5233 . . . . . 6  |-  ( z  =  B  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  B )
) )
53, 4opeq12d 3598 . . . . 5  |-  ( z  =  B  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
62, 5eqeq12d 2097 . . . 4  |-  ( z  =  B  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
76imbi2d 228 . . 3  |-  ( z  =  B  ->  (
( ph  ->  ( R `
 z )  = 
<. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >. )  <->  (
ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) ) )
8 fveq2 5229 . . . . 5  |-  ( z  =  (/)  ->  ( R `
 z )  =  ( R `  (/) ) )
9 fveq2 5229 . . . . . 6  |-  ( z  =  (/)  ->  ( G `
 z )  =  ( G `  (/) ) )
108fveq2d 5233 . . . . . 6  |-  ( z  =  (/)  ->  ( 2nd `  ( R `  z
) )  =  ( 2nd `  ( R `
 (/) ) ) )
119, 10opeq12d 3598 . . . . 5  |-  ( z  =  (/)  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
128, 11eqeq12d 2097 . . . 4  |-  ( z  =  (/)  ->  ( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `
 z ) )
>. 
<->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. ) )
13 fveq2 5229 . . . . 5  |-  ( z  =  v  ->  ( R `  z )  =  ( R `  v ) )
14 fveq2 5229 . . . . . 6  |-  ( z  =  v  ->  ( G `  z )  =  ( G `  v ) )
1513fveq2d 5233 . . . . . 6  |-  ( z  =  v  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  v )
) )
1614, 15opeq12d 3598 . . . . 5  |-  ( z  =  v  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. )
1713, 16eqeq12d 2097 . . . 4  |-  ( z  =  v  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  v
)  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. ) )
18 fveq2 5229 . . . . 5  |-  ( z  =  suc  v  -> 
( R `  z
)  =  ( R `
 suc  v )
)
19 fveq2 5229 . . . . . 6  |-  ( z  =  suc  v  -> 
( G `  z
)  =  ( G `
 suc  v )
)
2018fveq2d 5233 . . . . . 6  |-  ( z  =  suc  v  -> 
( 2nd `  ( R `  z )
)  =  ( 2nd `  ( R `  suc  v ) ) )
2119, 20opeq12d 3598 . . . . 5  |-  ( z  =  suc  v  ->  <. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >.  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
)
2218, 21eqeq12d 2097 . . . 4  |-  ( z  =  suc  v  -> 
( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
23 frecuzrdgrrn.2 . . . . . . 7  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
2423fveq1i 5230 . . . . . 6  |-  ( R `
 (/) )  =  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  (/) )
25 frec2uz.1 . . . . . . . 8  |-  ( ph  ->  C  e.  ZZ )
26 frecuzrdgrrn.a . . . . . . . 8  |-  ( ph  ->  A  e.  S )
27 opexg 4011 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  A  e.  S )  -> 
<. C ,  A >.  e. 
_V )
2825, 26, 27syl2anc 403 . . . . . . 7  |-  ( ph  -> 
<. C ,  A >.  e. 
_V )
29 frec0g 6066 . . . . . . 7  |-  ( <. C ,  A >.  e. 
_V  ->  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. ) `  (/) )  = 
<. C ,  A >. )
3028, 29syl 14 . . . . . 6  |-  ( ph  ->  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  (/) )  = 
<. C ,  A >. )
3124, 30syl5eq 2127 . . . . 5  |-  ( ph  ->  ( R `  (/) )  = 
<. C ,  A >. )
32 frec2uz.2 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
3325, 32frec2uz0d 9533 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
3431fveq2d 5233 . . . . . . 7  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  ( 2nd `  <. C ,  A >. )
)
35 uzid 8766 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
3625, 35syl 14 . . . . . . . 8  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
37 op2ndg 5829 . . . . . . . 8  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  ( 2nd `  <. C ,  A >. )  =  A )
3836, 26, 37syl2anc 403 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  A >. )  =  A )
3934, 38eqtrd 2115 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  A )
4033, 39opeq12d 3598 . . . . 5  |-  ( ph  -> 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>.  =  <. C ,  A >. )
4131, 40eqtr4d 2118 . . . 4  |-  ( ph  ->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
42 1st2nd2 5852 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4342adantl 271 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4443fveq2d 5233 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  om )  /\  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 )
>. ) )
45 df-ov 5566 . . . . . . . . . . . . . . 15  |-  ( ( 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 )
>. )
46 xp1st 5843 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
4746adantl 271 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
48 xp2nd 5844 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
4948adantl 271 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
50 peano2uz 8804 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  z )  e.  ( ZZ>= `  C
)  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
5147, 50syl 14 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z )  +  1 )  e.  (
ZZ>= `  C ) )
52 frecuzrdgrrn.f . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
5352ralrimivva 2448 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
5453ad2antrr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  A. x  e.  ( ZZ>= `  C ) A. y  e.  S  ( x F y )  e.  S )
55 oveq1 5570 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
5655eleq1d 2151 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
57 oveq2 5571 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
5857eleq1d 2151 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
5956, 58rspc2v 2721 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) )
6047, 49, 59syl2anc 403 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  v  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
) )
6154, 60mpd 13 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
)
62 opelxp 4420 . . . . . . . . . . . . . . . . 17  |-  ( <.
( ( 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
) )
6351, 61, 62sylanbrc 408 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.  e.  ( ( ZZ>= `  C
)  X.  S ) )
64 oveq1 5570 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
6564, 55opeq12d 3598 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
6657opeq2d 3597 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
67 eqid 2083 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
6865, 66, 67ovmpt2g 5686 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) >. )
6947, 49, 63, 68syl3anc 1170 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  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 ) ) >. )
7045, 69syl5eqr 2129 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  om )  /\  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
) ) >. )
7170, 63eqeltrd 2159 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  om )  /\  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 ) )
7244, 71eqeltrd 2159 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  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 ) )
7372ralrimiva 2439 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  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 ) )
7436adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  C  e.  ( ZZ>= `  C )
)
7526adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  A  e.  S )
76 opelxp 4420 . . . . . . . . . . . 12  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
7774, 75, 76sylanbrc 408 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
78 simpr 108 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  v  e.  om )
79 frecsuc 6076 . . . . . . . . . . 11  |-  ( ( 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 )  /\  v  e.  om )  ->  (frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  v )  =  ( ( 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 >. ) `  v
) ) )
8073, 77, 78, 79syl3anc 1170 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  om )  ->  (frec (
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  v )  =  ( ( 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 >. ) `  v
) ) )
8123fveq1i 5230 . . . . . . . . . 10  |-  ( R `
 suc  v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  v )
8223fveq1i 5230 . . . . . . . . . . 11  |-  ( R `
 v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  v
)
8382fveq2i 5232 . . . . . . . . . 10  |-  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  v
) )  =  ( ( 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 >. ) `  v
) )
8480, 81, 833eqtr4g 2140 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  suc  v )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  v ) ) )
8584adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  ( R `
 v ) ) )
86 fveq2 5229 . . . . . . . . 9  |-  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  ( R `
 v ) )  =  ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. ) )
87 df-ov 5566 . . . . . . . . . 10  |-  ( ( G `  v ) ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  v )
) )  =  ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 v ) ,  ( 2nd `  ( R `  v )
) >. )
8825adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  C  e.  ZZ )
8988, 32, 78frec2uzuzd 9536 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  (
ZZ>= `  C ) )
9025, 32, 26, 52, 23frecuzrdgrrn 9542 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  v )  e.  ( ( ZZ>= `  C )  X.  S ) )
91 xp2nd 5844 . . . . . . . . . . . 12  |-  ( ( R `  v )  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  ( R `  v
) )  e.  S
)
9290, 91syl 14 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( 2nd `  ( R `  v
) )  e.  S
)
93 peano2uz 8804 . . . . . . . . . . . . 13  |-  ( ( G `  v )  e.  ( ZZ>= `  C
)  ->  ( ( G `  v )  +  1 )  e.  ( ZZ>= `  C )
)
9489, 93syl 14 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )  +  1 )  e.  ( ZZ>= `  C )
)
9552caovclg 5704 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
9695adantlr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  om )  /\  (
z  e.  ( ZZ>= `  C )  /\  w  e.  S ) )  -> 
( z F w )  e.  S )
9796, 89, 92caovcld 5705 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
98 opelxp 4420 . . . . . . . . . . . 12  |-  ( <.
( ( G `  v )  +  1 ) ,  ( ( G `  v ) F ( 2nd `  ( R `  v )
) ) >.  e.  ( ( ZZ>= `  C )  X.  S )  <->  ( (
( G `  v
)  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
) )
9994, 97, 98sylanbrc 408 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
100 oveq1 5570 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w  +  1 )  =  ( ( G `
 v )  +  1 ) )
101 oveq1 5570 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w F z )  =  ( ( G `
 v ) F z ) )
102100, 101opeq12d 3598 . . . . . . . . . . . 12  |-  ( w  =  ( G `  v )  ->  <. (
w  +  1 ) ,  ( w F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F z ) >. )
103 oveq2 5571 . . . . . . . . . . . . 13  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  ( ( G `  v ) F z )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
104103opeq2d 3597 . . . . . . . . . . . 12  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
105 oveq1 5570 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x  +  1 )  =  ( w  + 
1 ) )
106 oveq1 5570 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F y )  =  ( w F y ) )
107105, 106opeq12d 3598 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( w  +  1 ) ,  ( w F y ) >. )
108 oveq2 5571 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  (
w F y )  =  ( w F z ) )
109108opeq2d 3597 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  <. (
w  +  1 ) ,  ( w F y ) >.  =  <. ( w  +  1 ) ,  ( w F z ) >. )
110107, 109cbvmpt2v 5635 . . . . . . . . . . . 12  |-  ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. )  =  (
w  e.  ( ZZ>= `  C ) ,  z  e.  S  |->  <. (
w  +  1 ) ,  ( w F z ) >. )
111102, 104, 110ovmpt2g 5686 . . . . . . . . . . 11  |-  ( ( ( G `  v
)  e.  ( ZZ>= `  C )  /\  ( 2nd `  ( R `  v ) )  e.  S  /\  <. (
( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >.  e.  ( ( ZZ>= `  C )  X.  S ) )  -> 
( ( G `  v ) ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  v )
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
11289, 92, 99, 111syl3anc 1170 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )
( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  v )
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
11387, 112syl5eqr 2129 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
11486, 113sylan9eqr 2137 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  v
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
11585, 114eqtrd 2115 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( ( G `
 v )  +  1 ) ,  ( ( G `  v
) F ( 2nd `  ( R `  v
) ) ) >.
)
11688, 32, 78frec2uzsucd 9535 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  suc  v )  =  ( ( G `  v )  +  1 ) )
117116adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( G `
 suc  v )  =  ( ( G `
 v )  +  1 ) )
118115fveq2d 5233 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( 2nd `  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
)
11988, 32, 78frec2uzzd 9534 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  ZZ )
120119peano2zd 8605 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
121120adantr 270 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
12297adantr 270 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
123 op2ndg 5829 . . . . . . . . . 10  |-  ( ( ( ( G `  v )  +  1 )  e.  ZZ  /\  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) )  e.  S )  ->  ( 2nd `  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )  =  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) )
124121, 122, 123syl2anc 403 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  <. ( ( G `
 v )  +  1 ) ,  ( ( G `  v
) F ( 2nd `  ( R `  v
) ) ) >.
)  =  ( ( G `  v ) F ( 2nd `  ( R `  v )
) ) )
125118, 124eqtrd 2115 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
126117, 125opeq12d 3598 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  <. ( G `  suc  v ) ,  ( 2nd `  ( R `  suc  v ) ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
127115, 126eqtr4d 2118 . . . . . 6  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. )
128127ex 113 . . . . 5  |-  ( (
ph  /\  v  e.  om )  ->  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
129128expcom 114 . . . 4  |-  ( v  e.  om  ->  ( ph  ->  ( ( R `
 v )  = 
<. ( G `  v
) ,  ( 2nd `  ( R `  v
) ) >.  ->  ( R `  suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. ) ) )
13012, 17, 22, 41, 129finds2 4370 . . 3  |-  ( z  e.  om  ->  ( ph  ->  ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. ) )
1317, 130vtoclga 2673 . 2  |-  ( B  e.  om  ->  ( ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
1321, 131mpcom 36 1  |-  ( ph  ->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2353   _Vcvv 2610   (/)c0 3267   <.cop 3419    |-> cmpt 3859   suc csuc 4148   omcom 4359    X. cxp 4389   ` cfv 4952  (class class class)co 5563    |-> cmpt2 5565   1stc1st 5816   2ndc2nd 5817  freccfrec 6059   1c1 7096    + caddc 7098   ZZcz 8484   ZZ>=cuz 8752
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753
This theorem is referenced by:  frecuzrdglem  9545  frecuzrdgtcl  9546  frecuzrdgsuc  9548
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