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Theorem algfx 11744
Description: If  F reaches a fixed point when the countdown function  C reaches  0,  F remains fixed after  N steps. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
algcvga.1  |-  F : S
--> S
algcvga.2  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
algcvga.3  |-  C : S
--> NN0
algcvga.4  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
algcvga.5  |-  N  =  ( C `  A
)
algfx.6  |-  ( z  e.  S  ->  (
( C `  z
)  =  0  -> 
( F `  z
)  =  z ) )
Assertion
Ref Expression
algfx  |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
Distinct variable groups:    z, C    z, F    z, R    z, S    z, K    z, N
Allowed substitution hint:    A( z)

Proof of Theorem algfx
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algcvga.5 . . . 4  |-  N  =  ( C `  A
)
2 algcvga.3 . . . . 5  |-  C : S
--> NN0
32ffvelrni 5554 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
41, 3eqeltrid 2226 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
54nn0zd 9183 . 2  |-  ( A  e.  S  ->  N  e.  ZZ )
6 uzval 9340 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { z  e.  ZZ  |  N  <_  z } )
76eleq2d 2209 . . . . . 6  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  <->  K  e.  { z  e.  ZZ  |  N  <_  z } ) )
87pm5.32i 449 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  K  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
9 fveqeq2 5430 . . . . . . 7  |-  ( m  =  N  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  N )  =  ( R `  N ) ) )
109imbi2d 229 . . . . . 6  |-  ( m  =  N  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  N
)  =  ( R `
 N ) ) ) )
11 fveqeq2 5430 . . . . . . 7  |-  ( m  =  k  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  k )  =  ( R `  N ) ) )
1211imbi2d 229 . . . . . 6  |-  ( m  =  k  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  k
)  =  ( R `
 N ) ) ) )
13 fveqeq2 5430 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
1413imbi2d 229 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  (
k  +  1 ) )  =  ( R `
 N ) ) ) )
15 fveqeq2 5430 . . . . . . 7  |-  ( m  =  K  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  K )  =  ( R `  N ) ) )
1615imbi2d 229 . . . . . 6  |-  ( m  =  K  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) ) )
17 eqidd 2140 . . . . . . 7  |-  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) )
1817a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) ) )
196eleq2d 2209 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
k  e.  ( ZZ>= `  N )  <->  k  e.  { z  e.  ZZ  |  N  <_  z } ) )
2019pm5.32i 449 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  k  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
21 eluznn0 9405 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
224, 21sylan 281 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
23 nn0uz 9372 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
24 algcvga.2 . . . . . . . . . . . . . . 15  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
25 0zd 9078 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  0  e.  ZZ )
26 id 19 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  A  e.  S )
27 algcvga.1 . . . . . . . . . . . . . . . 16  |-  F : S
--> S
2827a1i 9 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  F : S --> S )
2923, 24, 25, 26, 28algrp1 11738 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3022, 29syldan 280 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3123, 24, 25, 26, 28algrf 11737 . . . . . . . . . . . . . . . 16  |-  ( A  e.  S  ->  R : NN0 --> S )
3231ffvelrnda 5555 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
3322, 32syldan 280 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  k
)  e.  S )
34 algcvga.4 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
3527, 24, 2, 34, 1algcvga 11743 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  (
k  e.  ( ZZ>= `  N )  ->  ( C `  ( R `  k ) )  =  0 ) )
3635imp 123 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( C `  ( R `  k )
)  =  0 )
37 fveqeq2 5430 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( C `  z
)  =  0  <->  ( C `  ( R `  k ) )  =  0 ) )
38 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
39 id 19 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  z  =  ( R `  k ) )
4038, 39eqeq12d 2154 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( F `  z
)  =  z  <->  ( F `  ( R `  k
) )  =  ( R `  k ) ) )
4137, 40imbi12d 233 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R `  k )  ->  (
( ( C `  z )  =  0  ->  ( F `  z )  =  z )  <->  ( ( C `
 ( R `  k ) )  =  0  ->  ( F `  ( R `  k
) )  =  ( R `  k ) ) ) )
42 algfx.6 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  (
( C `  z
)  =  0  -> 
( F `  z
)  =  z ) )
4341, 42vtoclga 2752 . . . . . . . . . . . . . 14  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( R `  k )
)  =  0  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) ) )
4433, 36, 43sylc 62 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) )
4530, 44eqtrd 2172 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( R `
 k ) )
4645eqeq1d 2148 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  ( k  +  1 ) )  =  ( R `  N )  <-> 
( R `  k
)  =  ( R `
 N ) ) )
4746biimprd 157 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
4847expcom 115 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
4948adantl 275 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5020, 49sylbir 134 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  k  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5150a2d 26 . . . . . 6  |-  ( ( N  e.  ZZ  /\  k  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( ( A  e.  S  ->  ( R `  k )  =  ( R `  N ) )  ->  ( A  e.  S  ->  ( R `
 ( k  +  1 ) )  =  ( R `  N
) ) ) )
5210, 12, 14, 16, 18, 51uzind3 9176 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
538, 52sylbi 120 . . . 4  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
5453ex 114 . . 3  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( A  e.  S  ->  ( R `  K )  =  ( R `  N ) ) ) )
5554com3r 79 . 2  |-  ( A  e.  S  ->  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) ) )
565, 55mpd 13 1  |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2308   {crab 2420   {csn 3527   class class class wbr 3929    X. cxp 4537    o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   0cc0 7632   1c1 7633    + caddc 7635    < clt 7812    <_ cle 7813   NN0cn0 8989   ZZcz 9066   ZZ>=cuz 9338    seqcseq 10230
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 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  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 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-seqfrec 10231
This theorem is referenced by: (None)
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