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Theorem ialgfx 10274
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 } ) ,  S )
algcvga.3  |-  C : S
--> NN0
algcvga.4  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
algcvga.5  |-  N  =  ( C `  A
)
ialgcvga.s  |-  S  e.  V
algfx.6  |-  ( z  e.  S  ->  (
( C `  z
)  =  0  -> 
( F `  z
)  =  z ) )
Assertion
Ref Expression
ialgfx  |-  ( 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 hints:    A( z)    V( z)

Proof of Theorem ialgfx
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 5329 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
41, 3syl5eqel 2140 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
54nn0zd 8417 . 2  |-  ( A  e.  S  ->  N  e.  ZZ )
6 uzval 8571 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { z  e.  ZZ  |  N  <_  z } )
76eleq2d 2123 . . . . . 6  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  <->  K  e.  { z  e.  ZZ  |  N  <_  z } ) )
87pm5.32i 435 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  K  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
9 fveq2 5206 . . . . . . . 8  |-  ( m  =  N  ->  ( R `  m )  =  ( R `  N ) )
109eqeq1d 2064 . . . . . . 7  |-  ( m  =  N  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  N )  =  ( R `  N ) ) )
1110imbi2d 223 . . . . . 6  |-  ( m  =  N  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  N
)  =  ( R `
 N ) ) ) )
12 fveq2 5206 . . . . . . . 8  |-  ( m  =  k  ->  ( R `  m )  =  ( R `  k ) )
1312eqeq1d 2064 . . . . . . 7  |-  ( m  =  k  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  k )  =  ( R `  N ) ) )
1413imbi2d 223 . . . . . 6  |-  ( m  =  k  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  k
)  =  ( R `
 N ) ) ) )
15 fveq2 5206 . . . . . . . 8  |-  ( m  =  ( k  +  1 )  ->  ( R `  m )  =  ( R `  ( k  +  1 ) ) )
1615eqeq1d 2064 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
1716imbi2d 223 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  (
k  +  1 ) )  =  ( R `
 N ) ) ) )
18 fveq2 5206 . . . . . . . 8  |-  ( m  =  K  ->  ( R `  m )  =  ( R `  K ) )
1918eqeq1d 2064 . . . . . . 7  |-  ( m  =  K  ->  (
( R `  m
)  =  ( R `
 N )  <->  ( R `  K )  =  ( R `  N ) ) )
2019imbi2d 223 . . . . . 6  |-  ( m  =  K  ->  (
( A  e.  S  ->  ( R `  m
)  =  ( R `
 N ) )  <-> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) ) )
21 eqidd 2057 . . . . . . 7  |-  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) )
2221a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  ( A  e.  S  ->  ( R `  N )  =  ( R `  N ) ) )
236eleq2d 2123 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
k  e.  ( ZZ>= `  N )  <->  k  e.  { z  e.  ZZ  |  N  <_  z } ) )
2423pm5.32i 435 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  <->  ( N  e.  ZZ  /\  k  e. 
{ z  e.  ZZ  |  N  <_  z } ) )
25 eluznn0 8633 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
264, 25sylan 271 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
k  e.  NN0 )
27 nn0uz 8603 . . . . . . . . . . . . . . 15  |-  NN0  =  ( ZZ>= `  0 )
28 algcvga.2 . . . . . . . . . . . . . . 15  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
29 0zd 8314 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  0  e.  ZZ )
30 id 19 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  A  e.  S )
31 algcvga.1 . . . . . . . . . . . . . . . 16  |-  F : S
--> S
3231a1i 9 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  F : S --> S )
33 ialgcvga.s . . . . . . . . . . . . . . . 16  |-  S  e.  V
3433a1i 9 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  S  e.  V )
3527, 28, 29, 30, 32, 34ialgrp1 10268 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3626, 35syldan 270 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3727, 28, 29, 30, 32, 34ialgrf 10267 . . . . . . . . . . . . . . . 16  |-  ( A  e.  S  ->  R : NN0 --> S )
3837ffvelrnda 5330 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
3926, 38syldan 270 . . . . . . . . . . . . . 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 ) ) )
4131, 28, 2, 40, 1, 33ialgcvga 10273 . . . . . . . . . . . . . . 15  |-  ( A  e.  S  ->  (
k  e.  ( ZZ>= `  N )  ->  ( C `  ( R `  k ) )  =  0 ) )
4241imp 119 . . . . . . . . . . . . . 14  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( C `  ( R `  k )
)  =  0 )
43 fveq2 5206 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
4443eqeq1d 2064 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( C `  z
)  =  0  <->  ( C `  ( R `  k ) )  =  0 ) )
45 fveq2 5206 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
46 id 19 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( R `  k )  ->  z  =  ( R `  k ) )
4745, 46eqeq12d 2070 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( R `  k )  ->  (
( F `  z
)  =  z  <->  ( F `  ( R `  k
) )  =  ( R `  k ) ) )
4844, 47imbi12d 227 . . . . . . . . . . . . . . 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 2636 . . . . . . . . . . . . . 14  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( R `  k )
)  =  0  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) ) )
5139, 42, 50sylc 60 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( F `  ( R `  k )
)  =  ( R `
 k ) )
5236, 51eqtrd 2088 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( R `  (
k  +  1 ) )  =  ( R `
 k ) )
5352eqeq1d 2064 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  ( k  +  1 ) )  =  ( R `  N )  <-> 
( R `  k
)  =  ( R `
 N ) ) )
5453biimprd 151 . . . . . . . . . 10  |-  ( ( A  e.  S  /\  k  e.  ( ZZ>= `  N ) )  -> 
( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) )
5554expcom 113 . . . . . . . . 9  |-  ( k  e.  ( ZZ>= `  N
)  ->  ( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5655adantl 266 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  k  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5724, 56sylbir 129 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  k  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( ( R `  k )  =  ( R `  N )  ->  ( R `  ( k  +  1 ) )  =  ( R `  N ) ) ) )
5857a2d 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
) ) ) )
5911, 14, 17, 20, 22, 58uzind3 8410 . . . . 5  |-  ( ( N  e.  ZZ  /\  K  e.  { z  e.  ZZ  |  N  <_ 
z } )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
608, 59sylbi 118 . . . 4  |-  ( ( N  e.  ZZ  /\  K  e.  ( ZZ>= `  N ) )  -> 
( A  e.  S  ->  ( R `  K
)  =  ( R `
 N ) ) )
6160ex 112 . . 3  |-  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( A  e.  S  ->  ( R `  K )  =  ( R `  N ) ) ) )
6261com3r 77 . 2  |-  ( A  e.  S  ->  ( N  e.  ZZ  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) ) )
635, 62mpd 13 1  |-  ( A  e.  S  ->  ( K  e.  ( ZZ>= `  N )  ->  ( R `  K )  =  ( R `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409    =/= wne 2220   {crab 2327   {csn 3403   class class class wbr 3792    X. cxp 4371    o. ccom 4377   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   0cc0 6947   1c1 6948    + caddc 6950    < clt 7119    <_ cle 7120   NN0cn0 8239   ZZcz 8302   ZZ>=cuz 8569    seqcseq 9375
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 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376
This theorem is referenced by: (None)
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