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Theorem algfx 14220
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 6031 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
41, 3syl5eqel 2549 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
54nn0zd 10988 . 2  |-  ( A  e.  S  ->  N  e.  ZZ )
6 uzval 11108 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { z  e.  ZZ  |  N  <_  z } )
76eleq2d 2527 . . . . . 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 5872 . . . . . . . 8  |-  ( m  =  N  ->  ( R `  m )  =  ( R `  N ) )
109eqeq1d 2459 . . . . . . 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 5872 . . . . . . . 8  |-  ( m  =  k  ->  ( R `  m )  =  ( R `  k ) )
1312eqeq1d 2459 . . . . . . 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 5872 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( R `  m )  =  ( R `  ( k  +  1 ) ) )
1615eqeq1d 2459 . . . . . . 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 5872 . . . . . . . 8  |-  ( m  =  K  ->  ( R `  m )  =  ( R `  K ) )
1918eqeq1d 2459 . . . . . . 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 2458 . . . . . . 7  |-  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) )
2221a1i 11 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) ) )
236eleq2d 2527 . . . . . . . . 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 11176 . . . . . . . . . . . . . . 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 11140 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
28 algcvga.2 . . . . . . . . . . . . . . 15  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
29 0zd 10897 . . . . . . . . . . . . . . 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 14214 . . . . . . . . . . . . . 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 14213 . . . . . . . . . . . . . . . 16  |-  ( A  e.  S  ->  R : NN0 --> S )
3635ffvelrnda 6032 . . . . . . . . . . . . . . 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 14219 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
4241eqeq1d 2459 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( C `  z
)  =  0  <->  ( C `  ( R `  k ) )  =  0 ) )
43 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
44 id 22 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  z  =  ( R `  k ) )
4543, 44eqeq12d 2479 . . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( R `
 k ) )
5150eqeq1d 2459 . . . . . . . . . . 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 10977 . . . . 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 1395    e. wcel 1819    =/= wne 2652   {crab 2811   {csn 4032   class class class wbr 4456    X. cxp 5006    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645    <_ cle 9646   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106    seqcseq 12109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110
This theorem is referenced by: (None)
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