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Theorem ialginv 9886
Description: If  I is an invariant of  F, its value is unchanged after any number of iterations of  F. (Contributed by Paul Chapman, 31-Mar-2011.)
Hypotheses
Ref Expression
alginv.1  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
alginv.2  |-  F : S
--> S
alginv.3  |-  I  Fn  S
alginv.4  |-  ( x  e.  S  ->  (
I `  ( F `  x ) )  =  ( I `  x
) )
alginv.s  |-  S  e.  V
Assertion
Ref Expression
ialginv  |-  ( ( A  e.  S  /\  K  e.  NN0 )  -> 
( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) )
Distinct variable groups:    x, F    x, I    x, R    x, S
Allowed substitution hints:    A( x)    K( x)    V( x)

Proof of Theorem ialginv
Dummy variables  z  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5178 . . . . . 6  |-  ( z  =  0  ->  ( R `  z )  =  ( R ` 
0 ) )
21fveq2d 5182 . . . . 5  |-  ( z  =  0  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  0 )
) )
32eqeq1d 2048 . . . 4  |-  ( z  =  0  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  0
) )  =  ( I `  ( R `
 0 ) ) ) )
43imbi2d 219 . . 3  |-  ( z  =  0  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  0 )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
5 fveq2 5178 . . . . . 6  |-  ( z  =  k  ->  ( R `  z )  =  ( R `  k ) )
65fveq2d 5182 . . . . 5  |-  ( z  =  k  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  k )
) )
76eqeq1d 2048 . . . 4  |-  ( z  =  k  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
87imbi2d 219 . . 3  |-  ( z  =  k  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
9 fveq2 5178 . . . . . 6  |-  ( z  =  ( k  +  1 )  ->  ( R `  z )  =  ( R `  ( k  +  1 ) ) )
109fveq2d 5182 . . . . 5  |-  ( z  =  ( k  +  1 )  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  ( k  +  1 ) ) ) )
1110eqeq1d 2048 . . . 4  |-  ( z  =  ( k  +  1 )  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) )
1211imbi2d 219 . . 3  |-  ( z  =  ( k  +  1 )  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) ) )
13 fveq2 5178 . . . . . 6  |-  ( z  =  K  ->  ( R `  z )  =  ( R `  K ) )
1413fveq2d 5182 . . . . 5  |-  ( z  =  K  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  K )
) )
1514eqeq1d 2048 . . . 4  |-  ( z  =  K  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  K
) )  =  ( I `  ( R `
 0 ) ) ) )
1615imbi2d 219 . . 3  |-  ( z  =  K  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
17 eqidd 2041 . . 3  |-  ( A  e.  S  ->  (
I `  ( R `  0 ) )  =  ( I `  ( R `  0 ) ) )
18 nn0uz 8507 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 alginv.1 . . . . . . . . . 10  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
20 0zd 8257 . . . . . . . . . 10  |-  ( A  e.  S  ->  0  e.  ZZ )
21 id 19 . . . . . . . . . 10  |-  ( A  e.  S  ->  A  e.  S )
22 alginv.2 . . . . . . . . . . 11  |-  F : S
--> S
2322a1i 9 . . . . . . . . . 10  |-  ( A  e.  S  ->  F : S --> S )
24 alginv.s . . . . . . . . . . 11  |-  S  e.  V
2524a1i 9 . . . . . . . . . 10  |-  ( A  e.  S  ->  S  e.  V )
2618, 19, 20, 21, 23, 25ialgrp1 9885 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
2726fveq2d 5182 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( F `  ( R `  k ) ) ) )
2818, 19, 20, 21, 23, 25ialgrf 9884 . . . . . . . . . 10  |-  ( A  e.  S  ->  R : NN0 --> S )
2928ffvelrnda 5302 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
30 fveq2 5178 . . . . . . . . . . . 12  |-  ( x  =  ( R `  k )  ->  ( F `  x )  =  ( F `  ( R `  k ) ) )
3130fveq2d 5182 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  ( R `  k ) ) ) )
32 fveq2 5178 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  x )  =  ( I `  ( R `  k ) ) )
3331, 32eqeq12d 2054 . . . . . . . . . 10  |-  ( x  =  ( R `  k )  ->  (
( I `  ( F `  x )
)  =  ( I `
 x )  <->  ( I `  ( F `  ( R `  k )
) )  =  ( I `  ( R `
 k ) ) ) )
34 alginv.4 . . . . . . . . . 10  |-  ( x  e.  S  ->  (
I `  ( F `  x ) )  =  ( I `  x
) )
3533, 34vtoclga 2619 . . . . . . . . 9  |-  ( ( R `  k )  e.  S  ->  (
I `  ( F `  ( R `  k
) ) )  =  ( I `  ( R `  k )
) )
3629, 35syl 14 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( F `  ( R `  k ) ) )  =  ( I `  ( R `  k ) ) )
3727, 36eqtrd 2072 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R `  k ) ) )
3837eqeq1d 2048 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
3938biimprd 147 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  k ) )  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) )
4039expcom 109 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  S  ->  (
( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) ) )
4140a2d 23 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  S  -> 
( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) )  ->  ( A  e.  S  ->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) ) )
424, 8, 12, 16, 17, 41nn0ind 8352 . 2  |-  ( K  e.  NN0  ->  ( A  e.  S  ->  (
I `  ( R `  K ) )  =  ( I `  ( R `  0 )
) ) )
4342impcom 116 1  |-  ( ( A  e.  S  /\  K  e.  NN0 )  -> 
( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   {csn 3375    X. cxp 4343    o. ccom 4349    Fn wfn 4897   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   0cc0 6889   1c1 6890    + caddc 6892   NN0cn0 8181    seqcseq 9211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212
This theorem is referenced by: (None)
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