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Theorem algfx 12750
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 5664 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
41, 3syl5eqel 2367 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
54nn0zd 10115 . 2  |-  ( A  e.  S  ->  N  e.  ZZ )
6 uzval 10232 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { z  e.  ZZ  |  N  <_  z } )
76eleq2d 2350 . . . . . 6  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  <->  K  e.  { z  e.  ZZ  |  N  <_  z } ) )
87pm5.32i 618 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  K  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
9 fveq2 5525 . . . . . . . 8  |-  ( m  =  N  ->  ( R `  m )  =  ( R `  N ) )
109eqeq1d 2291 . . . . . . 7  |-  ( m  =  N  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  N )  =  ( R `  N ) ) )
1110imbi2d 307 . . . . . 6  |-  ( m  =  N  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  N
)  =  ( R `
 N ) ) ) )
12 fveq2 5525 . . . . . . . 8  |-  ( m  =  k  ->  ( R `  m )  =  ( R `  k ) )
1312eqeq1d 2291 . . . . . . 7  |-  ( m  =  k  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  k )  =  ( R `  N ) ) )
1413imbi2d 307 . . . . . 6  |-  ( m  =  k  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  k
)  =  ( R `
 N ) ) ) )
15 fveq2 5525 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( R `  m )  =  ( R `  ( k  +  1 ) ) )
1615eqeq1d 2291 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
1716imbi2d 307 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  (
k  +  1 ) )  =  ( R `
 N ) ) ) )
18 fveq2 5525 . . . . . . . 8  |-  ( m  =  K  ->  ( R `  m )  =  ( R `  K ) )
1918eqeq1d 2291 . . . . . . 7  |-  ( m  =  K  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  K )  =  ( R `  N ) ) )
2019imbi2d 307 . . . . . 6  |-  ( m  =  K  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) ) )
21 eqidd 2284 . . . . . . 7  |-  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) )
2221a1i 10 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) ) )
236eleq2d 2350 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
k  e.  ( ZZ>= `  N )  <->  k  e.  { z  e.  ZZ  |  N  <_  z } ) )
2423pm5.32i 618 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  k  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
25 eluznn0 10288 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
264, 25sylan 457 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
27 nn0uz 10262 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
28 algcvga.2 . . . . . . . . . . . . . . 15  |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
29 0z 10035 . . . . . . . . . . . . . . . 16  |-  0  e.  ZZ
3029a1i 10 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  0  e.  ZZ )
31 id 19 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  A  e.  S )
32 algcvga.1 . . . . . . . . . . . . . . . 16  |-  F : S
--> S
3332a1i 10 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  F : S --> S )
3427, 28, 30, 31, 33algrp1 12744 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3526, 34syldan 456 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3627, 28, 30, 31, 33algrf 12743 . . . . . . . . . . . . . . . 16  |-  ( A  e.  S  ->  R : NN0 --> S )
37 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( R : NN0 --> S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
3836, 37sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
3926, 38syldan 456 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  k
)  e.  S )
40 algcvga.4 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
4132, 28, 2, 40, 1algcvga 12749 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  (
k  e.  ( ZZ>= `  N )  ->  ( C `  ( R `  k ) )  =  0 ) )
4241imp 418 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( C `  ( R `  k )
)  =  0 )
43 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
4443eqeq1d 2291 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( C `  z
)  =  0  <->  ( C `  ( R `  k ) )  =  0 ) )
45 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
46 id 19 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  z  =  ( R `  k ) )
4745, 46eqeq12d 2297 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( F `  z
)  =  z  <->  ( F `  ( R `  k
) )  =  ( R `  k ) ) )
4844, 47imbi12d 311 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R `  k )  ->  (
( ( C `  z )  =  0  ->  ( F `  z )  =  z )  <->  ( ( C `
 ( R `  k ) )  =  0  ->  ( F `  ( R `  k
) )  =  ( R `  k ) ) ) )
49 algfx.6 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  (
( C `  z
)  =  0  -> 
( F `  z
)  =  z ) )
5048, 49vtoclga 2849 . . . . . . . . . . . . . 14  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( R `  k )
)  =  0  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) ) )
5139, 42, 50sylc 56 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) )
5235, 51eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( R `
 k ) )
5352eqeq1d 2291 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  ( k  +  1 ) )  =  ( R `  N )  <-> 
( R `  k
)  =  ( R `
 N ) ) )
5453biimprd 214 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
5554expcom 424 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5655adantl 452 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5724, 56sylbir 204 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  k  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5857a2d 23 . . . . . 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
) ) ) )
5911, 14, 17, 20, 22, 58uzind3 10105 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
608, 59sylbi 187 . . . 4  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
6160ex 423 . . 3  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( A  e.  S  ->  ( R `  K )  =  ( R `  N ) ) ) )
6261com3r 73 . 2  |-  ( A  e.  S  ->  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) ) )
635, 62mpd 14 1  |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   {csn 3640   class class class wbr 4023    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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