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Theorem algfx 13754
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 5841 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
41, 3syl5eqel 2526 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
54nn0zd 10744 . 2  |-  ( A  e.  S  ->  N  e.  ZZ )
6 uzval 10862 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { z  e.  ZZ  |  N  <_  z } )
76eleq2d 2509 . . . . . 6  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  <->  K  e.  { z  e.  ZZ  |  N  <_  z } ) )
87pm5.32i 637 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  K  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
9 fveq2 5690 . . . . . . . 8  |-  ( m  =  N  ->  ( R `  m )  =  ( R `  N ) )
109eqeq1d 2450 . . . . . . 7  |-  ( m  =  N  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  N )  =  ( R `  N ) ) )
1110imbi2d 316 . . . . . 6  |-  ( m  =  N  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  N
)  =  ( R `
 N ) ) ) )
12 fveq2 5690 . . . . . . . 8  |-  ( m  =  k  ->  ( R `  m )  =  ( R `  k ) )
1312eqeq1d 2450 . . . . . . 7  |-  ( m  =  k  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  k )  =  ( R `  N ) ) )
1413imbi2d 316 . . . . . 6  |-  ( m  =  k  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  k
)  =  ( R `
 N ) ) ) )
15 fveq2 5690 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( R `  m )  =  ( R `  ( k  +  1 ) ) )
1615eqeq1d 2450 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
1716imbi2d 316 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  (
k  +  1 ) )  =  ( R `
 N ) ) ) )
18 fveq2 5690 . . . . . . . 8  |-  ( m  =  K  ->  ( R `  m )  =  ( R `  K ) )
1918eqeq1d 2450 . . . . . . 7  |-  ( m  =  K  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  K )  =  ( R `  N ) ) )
2019imbi2d 316 . . . . . 6  |-  ( m  =  K  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) ) )
21 eqidd 2443 . . . . . . 7  |-  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) )
2221a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) ) )
236eleq2d 2509 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
k  e.  ( ZZ>= `  N )  <->  k  e.  { z  e.  ZZ  |  N  <_  z } ) )
2423pm5.32i 637 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  k  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
25 eluznn0 10923 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
264, 25sylan 471 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
27 nn0uz 10894 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
28 algcvga.2 . . . . . . . . . . . . . . 15  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
29 0zd 10657 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  0  e.  ZZ )
30 id 22 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  A  e.  S )
31 algcvga.1 . . . . . . . . . . . . . . . 16  |-  F : S
--> S
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  F : S --> S )
3327, 28, 29, 30, 32algrp1 13748 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3426, 33syldan 470 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3527, 28, 29, 30, 32algrf 13747 . . . . . . . . . . . . . . . 16  |-  ( A  e.  S  ->  R : NN0 --> S )
3635ffvelrnda 5842 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
3726, 36syldan 470 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  k
)  e.  S )
38 algcvga.4 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
3931, 28, 2, 38, 1algcvga 13753 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  (
k  e.  ( ZZ>= `  N )  ->  ( C `  ( R `  k ) )  =  0 ) )
4039imp 429 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( C `  ( R `  k )
)  =  0 )
41 fveq2 5690 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
4241eqeq1d 2450 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( C `  z
)  =  0  <->  ( C `  ( R `  k ) )  =  0 ) )
43 fveq2 5690 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
44 id 22 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  z  =  ( R `  k ) )
4543, 44eqeq12d 2456 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( F `  z
)  =  z  <->  ( F `  ( R `  k
) )  =  ( R `  k ) ) )
4642, 45imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R `  k )  ->  (
( ( C `  z )  =  0  ->  ( F `  z )  =  z )  <->  ( ( C `
 ( R `  k ) )  =  0  ->  ( F `  ( R `  k
) )  =  ( R `  k ) ) ) )
47 algfx.6 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  (
( C `  z
)  =  0  -> 
( F `  z
)  =  z ) )
4846, 47vtoclga 3035 . . . . . . . . . . . . . 14  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( R `  k )
)  =  0  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) ) )
4937, 40, 48sylc 60 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) )
5034, 49eqtrd 2474 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( R `
 k ) )
5150eqeq1d 2450 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  ( k  +  1 ) )  =  ( R `  N )  <-> 
( R `  k
)  =  ( R `
 N ) ) )
5251biimprd 223 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
5352expcom 435 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5453adantl 466 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5524, 54sylbir 213 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  k  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5655a2d 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
) ) ) )
5711, 14, 17, 20, 22, 56uzind3 10734 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
588, 57sylbi 195 . . . 4  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
5958ex 434 . . 3  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( A  e.  S  ->  ( R `  K )  =  ( R `  N ) ) ) )
6059com3r 79 . 2  |-  ( A  e.  S  ->  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) ) )
615, 60mpd 15 1  |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   {crab 2718   {csn 3876   class class class wbr 4291    X. cxp 4837    o. ccom 4843   -->wf 5413   ` cfv 5417  (class class class)co 6090   1stc1st 6574   0cc0 9281   1c1 9282    + caddc 9284    < clt 9417    <_ cle 9418   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860    seqcseq 11805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-seq 11806
This theorem is referenced by: (None)
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