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Theorem ialginv 10620
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 5230 . . . . . 6  |-  ( z  =  0  ->  ( R `  z )  =  ( R ` 
0 ) )
21fveq2d 5234 . . . . 5  |-  ( z  =  0  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  0 )
) )
32eqeq1d 2091 . . . 4  |-  ( z  =  0  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  0
) )  =  ( I `  ( R `
 0 ) ) ) )
43imbi2d 228 . . 3  |-  ( z  =  0  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  0 )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
5 fveq2 5230 . . . . . 6  |-  ( z  =  k  ->  ( R `  z )  =  ( R `  k ) )
65fveq2d 5234 . . . . 5  |-  ( z  =  k  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  k )
) )
76eqeq1d 2091 . . . 4  |-  ( z  =  k  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
87imbi2d 228 . . 3  |-  ( z  =  k  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
9 fveq2 5230 . . . . . 6  |-  ( z  =  ( k  +  1 )  ->  ( R `  z )  =  ( R `  ( k  +  1 ) ) )
109fveq2d 5234 . . . . 5  |-  ( z  =  ( k  +  1 )  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  ( k  +  1 ) ) ) )
1110eqeq1d 2091 . . . 4  |-  ( z  =  ( k  +  1 )  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) )
1211imbi2d 228 . . 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 5230 . . . . . 6  |-  ( z  =  K  ->  ( R `  z )  =  ( R `  K ) )
1413fveq2d 5234 . . . . 5  |-  ( z  =  K  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  K )
) )
1514eqeq1d 2091 . . . 4  |-  ( z  =  K  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  K
) )  =  ( I `  ( R `
 0 ) ) ) )
1615imbi2d 228 . . 3  |-  ( z  =  K  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
17 eqidd 2084 . . 3  |-  ( A  e.  S  ->  (
I `  ( R `  0 ) )  =  ( I `  ( R `  0 ) ) )
18 nn0uz 8770 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 alginv.1 . . . . . . . . . 10  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
20 0zd 8480 . . . . . . . . . 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 10619 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
2726fveq2d 5234 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( F `  ( R `  k ) ) ) )
2818, 19, 20, 21, 23, 25ialgrf 10618 . . . . . . . . . 10  |-  ( A  e.  S  ->  R : NN0 --> S )
2928ffvelrnda 5355 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
30 fveq2 5230 . . . . . . . . . . . 12  |-  ( x  =  ( R `  k )  ->  ( F `  x )  =  ( F `  ( R `  k ) ) )
3130fveq2d 5234 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  ( R `  k ) ) ) )
32 fveq2 5230 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  x )  =  ( I `  ( R `  k ) ) )
3331, 32eqeq12d 2097 . . . . . . . . . 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 2673 . . . . . . . . 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 2115 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R `  k ) ) )
3837eqeq1d 2091 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
3938biimprd 156 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  k ) )  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) )
4039expcom 114 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  S  ->  (
( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) ) )
4140a2d 26 . . 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 8578 . 2  |-  ( K  e.  NN0  ->  ( A  e.  S  ->  (
I `  ( R `  K ) )  =  ( I `  ( R `  0 )
) ) )
4342impcom 123 1  |-  ( ( A  e.  S  /\  K  e.  NN0 )  -> 
( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {csn 3417    X. cxp 4390    o. ccom 4396    Fn wfn 4948   -->wf 4949   ` cfv 4953  (class class class)co 5564   1stc1st 5817   0cc0 7079   1c1 7080    + caddc 7082   NN0cn0 8391    seqcseq 9557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-n0 8392  df-z 8469  df-uz 8737  df-iseq 9558
This theorem is referenced by:  eucialg  10632
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