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Theorem frec2uzrdg 10206
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 10196 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 5424 . . . . 5  |-  ( z  =  B  ->  ( R `  z )  =  ( R `  B ) )
3 fveq2 5424 . . . . . 6  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
42fveq2d 5428 . . . . . 6  |-  ( z  =  B  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  B )
) )
53, 4opeq12d 3716 . . . . 5  |-  ( z  =  B  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
62, 5eqeq12d 2154 . . . 4  |-  ( z  =  B  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
76imbi2d 229 . . 3  |-  ( z  =  B  ->  (
( ph  ->  ( R `
 z )  = 
<. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >. )  <->  (
ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) ) )
8 fveq2 5424 . . . . 5  |-  ( z  =  (/)  ->  ( R `
 z )  =  ( R `  (/) ) )
9 fveq2 5424 . . . . . 6  |-  ( z  =  (/)  ->  ( G `
 z )  =  ( G `  (/) ) )
108fveq2d 5428 . . . . . 6  |-  ( z  =  (/)  ->  ( 2nd `  ( R `  z
) )  =  ( 2nd `  ( R `
 (/) ) ) )
119, 10opeq12d 3716 . . . . 5  |-  ( z  =  (/)  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
128, 11eqeq12d 2154 . . . 4  |-  ( z  =  (/)  ->  ( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `
 z ) )
>. 
<->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. ) )
13 fveq2 5424 . . . . 5  |-  ( z  =  v  ->  ( R `  z )  =  ( R `  v ) )
14 fveq2 5424 . . . . . 6  |-  ( z  =  v  ->  ( G `  z )  =  ( G `  v ) )
1513fveq2d 5428 . . . . . 6  |-  ( z  =  v  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  v )
) )
1614, 15opeq12d 3716 . . . . 5  |-  ( z  =  v  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. )
1713, 16eqeq12d 2154 . . . 4  |-  ( z  =  v  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  v
)  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. ) )
18 fveq2 5424 . . . . 5  |-  ( z  =  suc  v  -> 
( R `  z
)  =  ( R `
 suc  v )
)
19 fveq2 5424 . . . . . 6  |-  ( z  =  suc  v  -> 
( G `  z
)  =  ( G `
 suc  v )
)
2018fveq2d 5428 . . . . . 6  |-  ( z  =  suc  v  -> 
( 2nd `  ( R `  z )
)  =  ( 2nd `  ( R `  suc  v ) ) )
2119, 20opeq12d 3716 . . . . 5  |-  ( z  =  suc  v  ->  <. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >.  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
)
2218, 21eqeq12d 2154 . . . 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 5425 . . . . . 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 4153 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  A  e.  S )  -> 
<. C ,  A >.  e. 
_V )
2825, 26, 27syl2anc 408 . . . . . . 7  |-  ( ph  -> 
<. C ,  A >.  e. 
_V )
29 frec0g 6297 . . . . . . 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 2184 . . . . 5  |-  ( ph  ->  ( R `  (/) )  = 
<. C ,  A >. )
32 frec2uz.2 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
3325, 32frec2uz0d 10196 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
3431fveq2d 5428 . . . . . . 7  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  ( 2nd `  <. C ,  A >. )
)
35 uzid 9359 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
3625, 35syl 14 . . . . . . . 8  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
37 op2ndg 6052 . . . . . . . 8  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  ( 2nd `  <. C ,  A >. )  =  A )
3836, 26, 37syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  A >. )  =  A )
3934, 38eqtrd 2172 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  A )
4033, 39opeq12d 3716 . . . . 5  |-  ( ph  -> 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>.  =  <. C ,  A >. )
4131, 40eqtr4d 2175 . . . 4  |-  ( ph  ->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
42 1st2nd2 6076 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4342adantl 275 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
4443fveq2d 5428 . . . . . . . . . . . . 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 5780 . . . . . . . . . . . . . . 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 6066 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
4746adantl 275 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 1st `  z )  e.  (
ZZ>= `  C ) )
48 xp2nd 6067 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  z )  e.  S
)
4948adantl 275 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  om )  /\  z  e.  ( ( ZZ>= `  C
)  X.  S ) )  ->  ( 2nd `  z )  e.  S
)
50 peano2uz 9400 . . . . . . . . . . . . . . . . . 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 2514 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. x  e.  (
ZZ>= `  C ) A. y  e.  S  (
x F y )  e.  S )
5453ad2antrr 479 . . . . . . . . . . . . . . . . . 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 5784 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  ( 1st `  z
)  ->  ( x F y )  =  ( ( 1st `  z
) F y ) )
5655eleq1d 2208 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  ( 1st `  z
)  ->  ( (
x F y )  e.  S  <->  ( ( 1st `  z ) F y )  e.  S
) )
57 oveq2 5785 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z ) F y )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
5857eleq1d 2208 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( 1st `  z
) F y )  e.  S  <->  ( ( 1st `  z ) F ( 2nd `  z
) )  e.  S
) )
5956, 58rspc2v 2802 . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . 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 4572 . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 5784 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  1 )  =  ( ( 1st `  z
)  +  1 ) )
6564, 55opeq12d 3716 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( 1st `  z
)  ->  <. ( x  +  1 ) ,  ( x F y ) >.  =  <. ( ( 1st `  z
)  +  1 ) ,  ( ( 1st `  z ) F y ) >. )
6657opeq2d 3715 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( 2nd `  z
)  ->  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F y )
>.  =  <. ( ( 1st `  z )  +  1 ) ,  ( ( 1st `  z
) F ( 2nd `  z ) ) >.
)
67 eqid 2139 . . . . . . . . . . . . . . . . 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, 67ovmpog 5908 . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . 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 2186 . . . . . . . . . . . . . 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 2216 . . . . . . . . . . . . 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 2216 . . . . . . . . . . . 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 2505 . . . . . . . . . . 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 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  C  e.  ( ZZ>= `  C )
)
7526adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  A  e.  S )
76 opelxp 4572 . . . . . . . . . . . 12  |-  ( <. C ,  A >.  e.  ( ( ZZ>= `  C
)  X.  S )  <-> 
( C  e.  (
ZZ>= `  C )  /\  A  e.  S )
)
7774, 75, 76sylanbrc 413 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. C ,  A >.  e.  ( (
ZZ>= `  C )  X.  S ) )
78 simpr 109 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  v  e.  om )
79 frecsuc 6307 . . . . . . . . . . 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 1216 . . . . . . . . . 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 5425 . . . . . . . . . 10  |-  ( R `
 suc  v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  v )
8223fveq1i 5425 . . . . . . . . . . 11  |-  ( R `
 v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  v
)
8382fveq2i 5427 . . . . . . . . . 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 2197 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  suc  v )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  v ) ) )
8584adantr 274 . . . . . . . 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 5424 . . . . . . . . 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 5780 . . . . . . . . . 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 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  C  e.  ZZ )
8988, 32, 78frec2uzuzd 10199 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  (
ZZ>= `  C ) )
9025, 32, 26, 52, 23frecuzrdgrrn 10205 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  v )  e.  ( ( ZZ>= `  C )  X.  S ) )
91 xp2nd 6067 . . . . . . . . . . . 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 9400 . . . . . . . . . . . . 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 5926 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
9695adantlr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  om )  /\  (
z  e.  ( ZZ>= `  C )  /\  w  e.  S ) )  -> 
( z F w )  e.  S )
9796, 89, 92caovcld 5927 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
98 opelxp 4572 . . . . . . . . . . . 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 413 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
100 oveq1 5784 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w  +  1 )  =  ( ( G `
 v )  +  1 ) )
101 oveq1 5784 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w F z )  =  ( ( G `
 v ) F z ) )
102100, 101opeq12d 3716 . . . . . . . . . . . 12  |-  ( w  =  ( G `  v )  ->  <. (
w  +  1 ) ,  ( w F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F z ) >. )
103 oveq2 5785 . . . . . . . . . . . . 13  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  ( ( G `  v ) F z )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
104103opeq2d 3715 . . . . . . . . . . . 12  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
105 oveq1 5784 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x  +  1 )  =  ( w  + 
1 ) )
106 oveq1 5784 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F y )  =  ( w F y ) )
107105, 106opeq12d 3716 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( w  +  1 ) ,  ( w F y ) >. )
108 oveq2 5785 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  (
w F y )  =  ( w F z ) )
109108opeq2d 3715 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  <. (
w  +  1 ) ,  ( w F y ) >.  =  <. ( w  +  1 ) ,  ( w F z ) >. )
110107, 109cbvmpov 5854 . . . . . . . . . . . 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, 110ovmpog 5908 . . . . . . . . . . 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 1216 . . . . . . . . . 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 2186 . . . . . . . . 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 2194 . . . . . . . 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 2172 . . . . . . 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 10198 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  suc  v )  =  ( ( G `  v )  +  1 ) )
117116adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( G `
 suc  v )  =  ( ( G `
 v )  +  1 ) )
118115fveq2d 5428 . . . . . . . . 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 10197 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  ZZ )
120119peano2zd 9195 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
121120adantr 274 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
12297adantr 274 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
123 op2ndg 6052 . . . . . . . . . 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 408 . . . . . . . . 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 2172 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
126117, 125opeq12d 3716 . . . . . . 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 2175 . . . . . 6  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. )
128127ex 114 . . . . 5  |-  ( (
ph  /\  v  e.  om )  ->  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
129128expcom 115 . . . 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 4518 . . 3  |-  ( z  e.  om  ->  ( ph  ->  ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. ) )
1317, 130vtoclga 2752 . 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 103    = wceq 1331    e. wcel 1480   A.wral 2416   _Vcvv 2686   (/)c0 3363   <.cop 3530    |-> cmpt 3992   suc csuc 4290   omcom 4507    X. cxp 4540   ` cfv 5126  (class class class)co 5777    e. cmpo 5779   1stc1st 6039   2ndc2nd 6040  freccfrec 6290   1c1 7640    + caddc 7642   ZZcz 9073   ZZ>=cuz 9345
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-addcom 7739  ax-addass 7741  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-0id 7747  ax-rnegex 7748  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-ltadd 7755
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-frec 6291  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-inn 8740  df-n0 8997  df-z 9074  df-uz 9346
This theorem is referenced by:  frecuzrdglem  10208  frecuzrdgtcl  10209  frecuzrdgsuc  10211
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