ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frecuzrdgsuc Unicode version

Theorem frecuzrdgsuc 9364
Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9348 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 )
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 >. )
frecuzrdgfn.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)    V( x, y)

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 265 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
3 frec2uz.2 . . . . . 6  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
4 uzrdg.s . . . . . . 7  |-  ( ph  ->  S  e.  V )
54adantr 265 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  S  e.  V )
6 uzrdg.a . . . . . . 7  |-  ( ph  ->  A  e.  S )
76adantr 265 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
8 uzrdg.f . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
98adantlr 454 . . . . . 6  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
10 uzrdg.2 . . . . . 6  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
11 peano2uz 8621 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
1211adantl 266 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
132, 3, 5, 7, 9, 10, 12frecuzrdglem 9360 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
14 frecuzrdgfn.3 . . . . . 6  |-  ( ph  ->  T  =  ran  R
)
1514adantr 265 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  T  =  ran  R )
1613, 15eleqtrrd 2133 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T
)
171, 3, 4, 6, 8, 10, 14frecuzrdgfn 9361 . . . . . . 7  |-  ( ph  ->  T  Fn  ( ZZ>= `  C ) )
18 fnfun 5023 . . . . . . 7  |-  ( T  Fn  ( ZZ>= `  C
)  ->  Fun  T )
1917, 18syl 14 . . . . . 6  |-  ( ph  ->  Fun  T )
20 funopfv 5240 . . . . . 6  |-  ( Fun 
T  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
2119, 20syl 14 . . . . 5  |-  ( ph  ->  ( <. ( B  + 
1 ) ,  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) )
>.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
2221adantr 265 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  T  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
2316, 22mpd 13 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) ) )
241, 3frec2uzf1od 9355 . . . . . . . . 9  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
25 f1ocnvdm 5448 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
2624, 25sylan 271 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  B )  e.  om )
272, 3, 26frec2uzsucd 9350 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
28 f1ocnvfv2 5445 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2924, 28sylan 271 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  =  B )
3029oveq1d 5554 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
3127, 30eqtrd 2088 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
32 peano2 4345 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  suc  ( `' G `  B )  e.  om )
3326, 32syl 14 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  suc  ( `' G `  B )  e.  om )
34 f1ocnvfv 5446 . . . . . . . 8  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  suc  ( `' G `  B )  e.  om )  ->  ( ( G `
 suc  ( `' G `  B )
)  =  ( B  +  1 )  -> 
( `' G `  ( B  +  1
) )  =  suc  ( `' G `  B ) ) )
3524, 34sylan 271 . . . . . . 7  |-  ( (
ph  /\  suc  ( `' G `  B )  e.  om )  -> 
( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) ) )
3633, 35syldan 270 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  -> 
( `' G `  ( B  +  1
) )  =  suc  ( `' G `  B ) ) )
3731, 36mpd 13 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) )
3837fveq2d 5209 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3938fveq2d 5209 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
4023, 39eqtrd 2088 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
41 zex 8310 . . . . . . . . . . 11  |-  ZZ  e.  _V
42 uzssz 8587 . . . . . . . . . . 11  |-  ( ZZ>= `  C )  C_  ZZ
4341, 42ssexi 3922 . . . . . . . . . 10  |-  ( ZZ>= `  C )  e.  _V
44 mpt2exga 5862 . . . . . . . . . 10  |-  ( ( ( ZZ>= `  C )  e.  _V  /\  S  e.  V )  ->  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  e.  _V )
4543, 44mpan 408 . . . . . . . . 9  |-  ( S  e.  V  ->  (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  e.  _V )
46 vex 2577 . . . . . . . . . 10  |-  z  e. 
_V
47 fvexg 5221 . . . . . . . . . 10  |-  ( ( ( 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 )
4846, 47mpan2 409 . . . . . . . . 9  |-  ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  e.  _V  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  _V )
495, 45, 483syl 17 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( (
x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  _V )
5049alrimiv 1770 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  A. z
( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) `  z )  e.  _V )
51 opelxp 4401 . . . . . . . . 9  |-  ( <. C ,  A >.  e.  ( ZZ  X.  S
)  <->  ( C  e.  ZZ  /\  A  e.  S ) )
521, 6, 51sylanbrc 402 . . . . . . . 8  |-  ( ph  -> 
<. C ,  A >.  e.  ( ZZ  X.  S
) )
5352adantr 265 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. C ,  A >.  e.  ( ZZ 
X.  S ) )
54 frecsuc 6021 . . . . . . 7  |-  ( ( A. z ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  z )  e.  _V  /\ 
<. C ,  A >.  e.  ( ZZ  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 ) ) ) )
5550, 53, 26, 54syl3anc 1146 . . . . . 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 ) ) ) )
5610fveq1i 5206 . . . . . 6  |-  ( R `
 suc  ( `' G `  B )
)  =  (frec ( ( x  e.  (
ZZ>= `  C ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. ) `  suc  ( `' G `  B ) )
5710fveq1i 5206 . . . . . . 7  |-  ( R `
 ( `' G `  B ) )  =  (frec ( ( x  e.  ( ZZ>= `  C
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. ) `  ( `' G `  B ) )
5857fveq2i 5208 . . . . . 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 ) ) )
5955, 56, 583eqtr4g 2113 . . . . 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 ) ) ) )
602, 3, 5, 7, 9, 10, 26frec2uzrdg 9358 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  = 
<. ( G `  ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
6160fveq2d 5209 . . . . . 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 )
) ) >. )
)
62 df-ov 5542 . . . . . 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 )
) ) >. )
6361, 62syl6eqr 2106 . . . . 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 )
) ) ) )
642, 3, 26frec2uzuzd 9351 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( G `  ( `' G `  B ) )  e.  ( ZZ>= `  C )
)
652, 3, 5, 7, 9, 10frecuzrdgrrn 9357 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( `' G `  B )  e.  om )  ->  ( R `  ( `' G `  B )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
6626, 65mpdan 406 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  B ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
67 xp2nd 5820 . . . . . . 7  |-  ( ( R `  ( `' G `  B ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
6866, 67syl 14 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  S
)
6930, 12eqeltrd 2130 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  e.  ( ZZ>= `  C
) )
709caovclg 5680 . . . . . . . 8  |-  ( ( ( ph  /\  B  e.  ( ZZ>= `  C )
)  /\  ( z  e.  ( ZZ>= `  C )  /\  w  e.  S
) )  ->  (
z F w )  e.  S )
7170, 64, 68caovcld 5681 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S )
72 opexg 3991 . . . . . . 7  |-  ( ( ( ( G `  ( `' G `  B ) )  +  1 )  e.  ( ZZ>= `  C
)  /\  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e.  S )  ->  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V )
7369, 71, 72syl2anc 397 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V )
74 oveq1 5546 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
75 oveq1 5546 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
7674, 75opeq12d 3584 . . . . . . 7  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
77 oveq2 5547 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
7877opeq2d 3583 . . . . . . 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 )
) ) ) >.
)
79 oveq1 5546 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
80 oveq1 5546 . . . . . . . . 9  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
8179, 80opeq12d 3584 . . . . . . . 8  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
82 oveq2 5547 . . . . . . . . 9  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
8382opeq2d 3583 . . . . . . . 8  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
8481, 83cbvmpt2v 5611 . . . . . . 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 ) >. )
8576, 78, 84ovmpt2g 5662 . . . . . 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.  _V )  ->  (
( 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 )
) ) ) >.
)
8664, 68, 73, 85syl3anc 1146 . . . . 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 )
) ) ) >.
)
8759, 63, 863eqtrd 2092 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( R `  suc  ( `' G `  B ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
8887fveq2d 5209 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( 2nd `  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
) )
89 op2ndg 5805 . . . 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 ) ) ) ) )
9069, 71, 89syl2anc 397 . . 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 ) ) ) ) )
9188, 90eqtrd 2088 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
92 simpr 107 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  B  e.  ( ZZ>= `  C )
)
932, 3, 5, 7, 9, 10, 92frecuzrdglem 9360 . . . . . 6  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
9493, 15eleqtrrd 2133 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T
)
95 funopfv 5240 . . . . . . 7  |-  ( Fun 
T  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T  ->  ( T `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
9619, 95syl 14 . . . . . 6  |-  ( ph  ->  ( <. B ,  ( 2nd `  ( R `
 ( `' G `  B ) ) )
>.  e.  T  ->  ( T `  B )  =  ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
9796adantr 265 . . . . 5  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  T  ->  ( T `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
9894, 97mpd 13 . . . 4  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
9998eqcomd 2061 . . 3  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( T `  B ) )
10029, 99oveq12d 5557 . 2  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( T `  B ) ) )
10140, 91, 1003eqtrd 2092 1  |-  ( (
ph  /\  B  e.  ( ZZ>= `  C )
)  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3405    |-> cmpt 3845   suc csuc 4129   omcom 4340    X. cxp 4370   `'ccnv 4371   ran crn 4373   Fun wfun 4923    Fn wfn 4924   -1-1-onto->wf1o 4928   ` 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:  iseqp1  9388
  Copyright terms: Public domain W3C validator