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Theorem frec2uzrdg 9353
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 9343 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 )
uzrdg.s  |-  ( ph  ->  S  e.  V )
uzrdg.a  |-  ( ph  ->  A  e.  S )
uzrdg.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
uzrdg.2  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
uzrdg.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)    V( x, y)

Proof of Theorem frec2uzrdg
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzrdg.b . 2  |-  ( ph  ->  B  e.  om )
2 fveq2 5205 . . . . 5  |-  ( z  =  B  ->  ( R `  z )  =  ( R `  B ) )
3 fveq2 5205 . . . . . 6  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
42fveq2d 5209 . . . . . 6  |-  ( z  =  B  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  B )
) )
53, 4opeq12d 3584 . . . . 5  |-  ( z  =  B  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
62, 5eqeq12d 2070 . . . 4  |-  ( z  =  B  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
76imbi2d 223 . . 3  |-  ( z  =  B  ->  (
( ph  ->  ( R `
 z )  = 
<. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >. )  <->  (
ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) ) )
8 fveq2 5205 . . . . 5  |-  ( z  =  (/)  ->  ( R `
 z )  =  ( R `  (/) ) )
9 fveq2 5205 . . . . . 6  |-  ( z  =  (/)  ->  ( G `
 z )  =  ( G `  (/) ) )
108fveq2d 5209 . . . . . 6  |-  ( z  =  (/)  ->  ( 2nd `  ( R `  z
) )  =  ( 2nd `  ( R `
 (/) ) ) )
119, 10opeq12d 3584 . . . . 5  |-  ( z  =  (/)  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
128, 11eqeq12d 2070 . . . 4  |-  ( z  =  (/)  ->  ( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `
 z ) )
>. 
<->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. ) )
13 fveq2 5205 . . . . 5  |-  ( z  =  v  ->  ( R `  z )  =  ( R `  v ) )
14 fveq2 5205 . . . . . 6  |-  ( z  =  v  ->  ( G `  z )  =  ( G `  v ) )
1513fveq2d 5209 . . . . . 6  |-  ( z  =  v  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  v )
) )
1614, 15opeq12d 3584 . . . . 5  |-  ( z  =  v  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. )
1713, 16eqeq12d 2070 . . . 4  |-  ( z  =  v  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  v
)  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. ) )
18 fveq2 5205 . . . . 5  |-  ( z  =  suc  v  -> 
( R `  z
)  =  ( R `
 suc  v )
)
19 fveq2 5205 . . . . . 6  |-  ( z  =  suc  v  -> 
( G `  z
)  =  ( G `
 suc  v )
)
2018fveq2d 5209 . . . . . 6  |-  ( z  =  suc  v  -> 
( 2nd `  ( R `  z )
)  =  ( 2nd `  ( R `  suc  v ) ) )
2119, 20opeq12d 3584 . . . . 5  |-  ( z  =  suc  v  ->  <. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >.  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
)
2218, 21eqeq12d 2070 . . . 4  |-  ( z  =  suc  v  -> 
( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
23 uzrdg.2 . . . . . . 7  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
2423fveq1i 5206 . . . . . 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 uzrdg.a . . . . . . . 8  |-  ( ph  ->  A  e.  S )
27 opexg 3991 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  A  e.  S )  -> 
<. C ,  A >.  e. 
_V )
2825, 26, 27syl2anc 397 . . . . . . 7  |-  ( ph  -> 
<. C ,  A >.  e. 
_V )
29 frec0g 6013 . . . . . . 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 2100 . . . . 5  |-  ( ph  ->  ( R `  (/) )  = 
<. C ,  A >. )
32 frec2uz.2 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
3325, 32frec2uz0d 9343 . . . . . 6  |-  ( ph  ->  ( G `  (/) )  =  C )
3431fveq2d 5209 . . . . . . 7  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  ( 2nd `  <. C ,  A >. )
)
35 uzid 8582 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  ( ZZ>= `  C )
)
3625, 35syl 14 . . . . . . . 8  |-  ( ph  ->  C  e.  ( ZZ>= `  C ) )
37 op2ndg 5805 . . . . . . . 8  |-  ( ( C  e.  ( ZZ>= `  C )  /\  A  e.  S )  ->  ( 2nd `  <. C ,  A >. )  =  A )
3836, 26, 37syl2anc 397 . . . . . . 7  |-  ( ph  ->  ( 2nd `  <. C ,  A >. )  =  A )
3934, 38eqtrd 2088 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( R `  (/) ) )  =  A )
4033, 39opeq12d 3584 . . . . 5  |-  ( ph  -> 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>.  =  <. C ,  A >. )
4131, 40eqtr4d 2091 . . . 4  |-  ( ph  ->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
42 zex 8310 . . . . . . . . . . . . . . . 16  |-  ZZ  e.  _V
43 uzssz 8587 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  C )  C_  ZZ
4442, 43ssexi 3922 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  C )  e.  _V
4544a1i 9 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  om )  ->  ( ZZ>= `  C )  e.  _V )
46 uzrdg.s . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  e.  V )
4746adantr 265 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  v  e.  om )  ->  S  e.  V )
48 mpt2exga 5862 . . . . . . . . . . . . . 14  |-  ( ( ( ZZ>= `  C )  e.  _V  /\  S  e.  V )  ->  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  e.  _V )
4945, 47, 48syl2anc 397 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  om )  ->  ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
)  e.  _V )
50 vex 2577 . . . . . . . . . . . . . 14  |-  z  e. 
_V
5150a1i 9 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  om )  ->  z  e.  _V )
52 fvexg 5221 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
)  e.  _V  /\  z  e.  _V )  ->  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  _V )
5349, 51, 52syl2anc 397 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  _V )
5453alrimiv 1770 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  A. z
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  _V )
5528adantr 265 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. C ,  A >.  e.  _V )
56 simpr 107 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  v  e.  om )
57 frecsuc 6021 . . . . . . . . . . 11  |-  ( ( A. z ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  _V  /\ 
<. C ,  A >.  e. 
_V  /\  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
) ) )
5854, 55, 56, 57syl3anc 1146 . . . . . . . . . 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
) ) )
5923fveq1i 5206 . . . . . . . . . 10  |-  ( R `
 suc  v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  suc  v )
6023fveq1i 5206 . . . . . . . . . . 11  |-  ( R `
 v )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  v
)
6160fveq2i 5208 . . . . . . . . . 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
) )
6258, 59, 613eqtr4g 2113 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  suc  v )  =  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  v ) ) )
6362adantr 265 . . . . . . . 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 ) ) )
64 fveq2 5205 . . . . . . . . 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 )
) >. ) )
65 df-ov 5542 . . . . . . . . . 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 )
) >. )
6625adantr 265 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  C  e.  ZZ )
6766, 32, 56frec2uzuzd 9346 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  (
ZZ>= `  C ) )
68 uzrdg.f . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
6925, 32, 46, 26, 68, 23frecuzrdgrrn 9352 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( R `  v )  e.  ( ( ZZ>= `  C )  X.  S ) )
70 xp2nd 5820 . . . . . . . . . . . 12  |-  ( ( R `  v )  e.  ( ( ZZ>= `  C )  X.  S
)  ->  ( 2nd `  ( R `  v
) )  e.  S
)
7169, 70syl 14 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( 2nd `  ( R `  v
) )  e.  S
)
72 peano2uz 8621 . . . . . . . . . . . . 13  |-  ( ( G `  v )  e.  ( ZZ>= `  C
)  ->  ( ( G `  v )  +  1 )  e.  ( ZZ>= `  C )
)
7367, 72syl 14 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )  +  1 )  e.  ( ZZ>= `  C )
)
7468caovclg 5680 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
7574adantlr 454 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  om )  /\  (
z  e.  ( ZZ>= `  C )  /\  w  e.  S ) )  -> 
( z F w )  e.  S )
7675, 67, 71caovcld 5681 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
77 opelxp 4401 . . . . . . . . . . . 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
) )
7873, 76, 77sylanbrc 402 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) >.  e.  (
( ZZ>= `  C )  X.  S ) )
79 oveq1 5546 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w  +  1 )  =  ( ( G `
 v )  +  1 ) )
80 oveq1 5546 . . . . . . . . . . . . 13  |-  ( w  =  ( G `  v )  ->  (
w F z )  =  ( ( G `
 v ) F z ) )
8179, 80opeq12d 3584 . . . . . . . . . . . 12  |-  ( w  =  ( G `  v )  ->  <. (
w  +  1 ) ,  ( w F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F z ) >. )
82 oveq2 5547 . . . . . . . . . . . . 13  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  ( ( G `  v ) F z )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
8382opeq2d 3583 . . . . . . . . . . . 12  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
84 oveq1 5546 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x  +  1 )  =  ( w  + 
1 ) )
85 oveq1 5546 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F y )  =  ( w F y ) )
8684, 85opeq12d 3584 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( w  +  1 ) ,  ( w F y ) >. )
87 oveq2 5547 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  (
w F y )  =  ( w F z ) )
8887opeq2d 3583 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  <. (
w  +  1 ) ,  ( w F y ) >.  =  <. ( w  +  1 ) ,  ( w F z ) >. )
8986, 88cbvmpt2v 5611 . . . . . . . . . . . 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 ) >. )
9081, 83, 89ovmpt2g 5662 . . . . . . . . . . 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 )
) ) >. )
9167, 71, 78, 90syl3anc 1146 . . . . . . . . . 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 )
) ) >. )
9265, 91syl5eqr 2102 . . . . . . . . 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 )
) ) >. )
9364, 92sylan9eqr 2110 . . . . . . . 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 )
) ) >. )
9463, 93eqtrd 2088 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( ( G `
 v )  +  1 ) ,  ( ( G `  v
) F ( 2nd `  ( R `  v
) ) ) >.
)
9566, 32, 56frec2uzsucd 9345 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  suc  v )  =  ( ( G `  v )  +  1 ) )
9695adantr 265 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( G `
 suc  v )  =  ( ( G `
 v )  +  1 ) )
9794fveq2d 5209 . . . . . . . . 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 )
) ) >. )
)
9866, 32, 56frec2uzzd 9344 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  om )  ->  ( G `  v )  e.  ZZ )
9998peano2zd 8421 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  om )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
10099adantr 265 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v )  +  1 )  e.  ZZ )
10176adantr 265 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  S
)
102 op2ndg 5805 . . . . . . . . . 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 )
) ) )
103100, 101, 102syl2anc 397 . . . . . . . . 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 )
) ) )
10497, 103eqtrd 2088 . . . . . . . 8  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
10596, 104opeq12d 3584 . . . . . . 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 )
) ) >. )
10694, 105eqtr4d 2091 . . . . . 6  |-  ( ( ( ph  /\  v  e.  om )  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. )
107106ex 112 . . . . 5  |-  ( (
ph  /\  v  e.  om )  ->  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
108107expcom 113 . . . 4  |-  ( v  e.  om  ->  ( ph  ->  ( ( R `
 v )  = 
<. ( G `  v
) ,  ( 2nd `  ( R `  v
) ) >.  ->  ( R `  suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. ) ) )
10912, 17, 22, 41, 108finds2 4351 . . 3  |-  ( z  e.  om  ->  ( ph  ->  ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. ) )
1107, 109vtoclga 2636 . 2  |-  ( B  e.  om  ->  ( ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
1111, 110mpcom 36 1  |-  ( ph  ->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574   (/)c0 3251   <.cop 3405    |-> cmpt 3845   suc csuc 4129   omcom 4340    X. cxp 4370   ` cfv 4929  (class class class)co 5539    |-> cmpt2 5541   2ndc2nd 5793  freccfrec 6007   1c1 6947    + caddc 6949   ZZcz 8301   ZZ>=cuz 8568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338  ax-cnex 7032  ax-resscn 7033  ax-1cn 7034  ax-1re 7035  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-addcom 7041  ax-addass 7043  ax-distr 7045  ax-i2m1 7046  ax-0id 7049  ax-rnegex 7050  ax-cnre 7052  ax-pre-ltirr 7053  ax-pre-ltwlin 7054  ax-pre-lttrn 7055  ax-pre-ltadd 7057
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-riota 5495  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-frec 6008  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873  df-1r 6874  df-0 6953  df-1 6954  df-r 6956  df-lt 6959  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-sub 7246  df-neg 7247  df-inn 7990  df-n0 8239  df-z 8302  df-uz 8569
This theorem is referenced by:  frecuzrdglem  9355  frecuzrdgfn  9356  frecuzrdgsuc  9359
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