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Theorem alginv 12745
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 5525 . . . . . 6  |-  ( z  =  0  ->  ( R `  z )  =  ( R ` 
0 ) )
21fveq2d 5529 . . . . 5  |-  ( z  =  0  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  0 )
) )
32eqeq1d 2291 . . . 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 5525 . . . . . 6  |-  ( z  =  k  ->  ( R `  z )  =  ( R `  k ) )
65fveq2d 5529 . . . . 5  |-  ( z  =  k  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  k )
) )
76eqeq1d 2291 . . . 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 5525 . . . . . 6  |-  ( z  =  ( k  +  1 )  ->  ( R `  z )  =  ( R `  ( k  +  1 ) ) )
109fveq2d 5529 . . . . 5  |-  ( z  =  ( k  +  1 )  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  ( k  +  1 ) ) ) )
1110eqeq1d 2291 . . . 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 5525 . . . . . 6  |-  ( z  =  K  ->  ( R `  z )  =  ( R `  K ) )
1413fveq2d 5529 . . . . 5  |-  ( z  =  K  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  K )
) )
1514eqeq1d 2291 . . . 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 2284 . . 3  |-  ( A  e.  S  ->  (
I `  ( R `  0 ) )  =  ( I `  ( R `  0 ) ) )
18 nn0uz 10262 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 alginv.1 . . . . . . . . . 10  |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
20 0z 10035 . . . . . . . . . . 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 12744 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
2625fveq2d 5529 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( F `  ( R `  k ) ) ) )
2718, 19, 21, 22, 24algrf 12743 . . . . . . . . . 10  |-  ( A  e.  S  ->  R : NN0 --> S )
28 ffvelrn 5663 . . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( x  =  ( R `  k )  ->  ( F `  x )  =  ( F `  ( R `  k ) ) )
3130fveq2d 5529 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  ( R `  k ) ) ) )
32 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  x )  =  ( I `  ( R `  k ) ) )
3331, 32eqeq12d 2297 . . . . . . . . . 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 2849 . . . . . . . . 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 2315 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R `  k ) ) )
3837eqeq1d 2291 . . . . . 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 10108 . 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 1623    e. wcel 1684   {csn 3640    X. cxp 4687    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965   ZZcz 10024    seq cseq 11046
This theorem is referenced by:  eucalg  12757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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