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Theorem alginv 12761
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 } ) )
alginv.2  |-  F : S
--> S
alginv.3  |-  I  Fn  S
alginv.4  |-  ( x  e.  S  ->  (
I `  ( F `  x ) )  =  ( I `  x
) )
Assertion
Ref Expression
alginv  |-  ( ( 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)

Proof of Theorem alginv
Dummy variables  z 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . . 6  |-  ( z  =  0  ->  ( R `  z )  =  ( R ` 
0 ) )
21fveq2d 5545 . . . . 5  |-  ( z  =  0  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  0 )
) )
32eqeq1d 2304 . . . 4  |-  ( z  =  0  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  0
) )  =  ( I `  ( R `
 0 ) ) ) )
43imbi2d 307 . . 3  |-  ( z  =  0  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  0 )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
5 fveq2 5541 . . . . . 6  |-  ( z  =  k  ->  ( R `  z )  =  ( R `  k ) )
65fveq2d 5545 . . . . 5  |-  ( z  =  k  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  k )
) )
76eqeq1d 2304 . . . 4  |-  ( z  =  k  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
87imbi2d 307 . . 3  |-  ( z  =  k  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
9 fveq2 5541 . . . . . 6  |-  ( z  =  ( k  +  1 )  ->  ( R `  z )  =  ( R `  ( k  +  1 ) ) )
109fveq2d 5545 . . . . 5  |-  ( z  =  ( k  +  1 )  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  ( k  +  1 ) ) ) )
1110eqeq1d 2304 . . . 4  |-  ( z  =  ( k  +  1 )  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) )
1211imbi2d 307 . . 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 5541 . . . . . 6  |-  ( z  =  K  ->  ( R `  z )  =  ( R `  K ) )
1413fveq2d 5545 . . . . 5  |-  ( z  =  K  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  K )
) )
1514eqeq1d 2304 . . . 4  |-  ( z  =  K  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  K
) )  =  ( I `  ( R `
 0 ) ) ) )
1615imbi2d 307 . . 3  |-  ( z  =  K  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
17 eqidd 2297 . . 3  |-  ( A  e.  S  ->  (
I `  ( R `  0 ) )  =  ( I `  ( R `  0 ) ) )
18 nn0uz 10278 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 alginv.1 . . . . . . . . . 10  |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
20 0z 10051 . . . . . . . . . . 11  |-  0  e.  ZZ
2120a1i 10 . . . . . . . . . 10  |-  ( A  e.  S  ->  0  e.  ZZ )
22 id 19 . . . . . . . . . 10  |-  ( A  e.  S  ->  A  e.  S )
23 alginv.2 . . . . . . . . . . 11  |-  F : S
--> S
2423a1i 10 . . . . . . . . . 10  |-  ( A  e.  S  ->  F : S --> S )
2518, 19, 21, 22, 24algrp1 12760 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
2625fveq2d 5545 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( F `  ( R `  k ) ) ) )
2718, 19, 21, 22, 24algrf 12759 . . . . . . . . . 10  |-  ( A  e.  S  ->  R : NN0 --> S )
28 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( R : NN0 --> S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
2927, 28sylan 457 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
30 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  ( R `  k )  ->  ( F `  x )  =  ( F `  ( R `  k ) ) )
3130fveq2d 5545 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  ( R `  k ) ) ) )
32 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  x )  =  ( I `  ( R `  k ) ) )
3331, 32eqeq12d 2310 . . . . . . . . . 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 2862 . . . . . . . . 9  |-  ( ( R `  k )  e.  S  ->  (
I `  ( F `  ( R `  k
) ) )  =  ( I `  ( R `  k )
) )
3629, 35syl 15 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( F `  ( R `  k ) ) )  =  ( I `  ( R `  k ) ) )
3726, 36eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R `  k ) ) )
3837eqeq1d 2304 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
3938biimprd 214 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  k ) )  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) )
4039expcom 424 . . . 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 10124 . 2  |-  ( K  e.  NN0  ->  ( A  e.  S  ->  (
I `  ( R `  K ) )  =  ( I `  ( R `  0 )
) ) )
4342impcom 419 1  |-  ( ( A  e.  S  /\  K  e.  NN0 )  -> 
( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653    X. cxp 4703    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   0cc0 8753   1c1 8754    + caddc 8756   NN0cn0 9981   ZZcz 10040    seq cseq 11062
This theorem is referenced by:  eucalg  12773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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