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Theorem alginv 13066
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 5728 . . . . . 6  |-  ( z  =  0  ->  ( R `  z )  =  ( R ` 
0 ) )
21fveq2d 5732 . . . . 5  |-  ( z  =  0  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  0 )
) )
32eqeq1d 2444 . . . 4  |-  ( z  =  0  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  0
) )  =  ( I `  ( R `
 0 ) ) ) )
43imbi2d 308 . . 3  |-  ( z  =  0  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  0 )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
5 fveq2 5728 . . . . . 6  |-  ( z  =  k  ->  ( R `  z )  =  ( R `  k ) )
65fveq2d 5732 . . . . 5  |-  ( z  =  k  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  k )
) )
76eqeq1d 2444 . . . 4  |-  ( z  =  k  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
87imbi2d 308 . . 3  |-  ( z  =  k  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
9 fveq2 5728 . . . . . 6  |-  ( z  =  ( k  +  1 )  ->  ( R `  z )  =  ( R `  ( k  +  1 ) ) )
109fveq2d 5732 . . . . 5  |-  ( z  =  ( k  +  1 )  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  ( k  +  1 ) ) ) )
1110eqeq1d 2444 . . . 4  |-  ( z  =  ( k  +  1 )  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) )
1211imbi2d 308 . . 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 5728 . . . . . 6  |-  ( z  =  K  ->  ( R `  z )  =  ( R `  K ) )
1413fveq2d 5732 . . . . 5  |-  ( z  =  K  ->  (
I `  ( R `  z ) )  =  ( I `  ( R `  K )
) )
1514eqeq1d 2444 . . . 4  |-  ( z  =  K  ->  (
( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  K
) )  =  ( I `  ( R `
 0 ) ) ) )
1615imbi2d 308 . . 3  |-  ( z  =  K  ->  (
( A  e.  S  ->  ( I `  ( R `  z )
)  =  ( I `
 ( R ` 
0 ) ) )  <-> 
( A  e.  S  ->  ( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) ) ) )
17 eqidd 2437 . . 3  |-  ( A  e.  S  ->  (
I `  ( R `  0 ) )  =  ( I `  ( R `  0 ) ) )
18 nn0uz 10520 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 alginv.1 . . . . . . . . . 10  |-  R  =  seq  0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
20 0z 10293 . . . . . . . . . . 11  |-  0  e.  ZZ
2120a1i 11 . . . . . . . . . 10  |-  ( A  e.  S  ->  0  e.  ZZ )
22 id 20 . . . . . . . . . 10  |-  ( A  e.  S  ->  A  e.  S )
23 alginv.2 . . . . . . . . . . 11  |-  F : S
--> S
2423a1i 11 . . . . . . . . . 10  |-  ( A  e.  S  ->  F : S --> S )
2518, 19, 21, 22, 24algrp1 13065 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
2625fveq2d 5732 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( F `  ( R `  k ) ) ) )
2718, 19, 21, 22, 24algrf 13064 . . . . . . . . . 10  |-  ( A  e.  S  ->  R : NN0 --> S )
2827ffvelrnda 5870 . . . . . . . . 9  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
29 fveq2 5728 . . . . . . . . . . . 12  |-  ( x  =  ( R `  k )  ->  ( F `  x )  =  ( F `  ( R `  k ) ) )
3029fveq2d 5732 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  ( R `  k ) ) ) )
31 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  ( R `  k )  ->  (
I `  x )  =  ( I `  ( R `  k ) ) )
3230, 31eqeq12d 2450 . . . . . . . . . 10  |-  ( x  =  ( R `  k )  ->  (
( I `  ( F `  x )
)  =  ( I `
 x )  <->  ( I `  ( F `  ( R `  k )
) )  =  ( I `  ( R `
 k ) ) ) )
33 alginv.4 . . . . . . . . . 10  |-  ( x  e.  S  ->  (
I `  ( F `  x ) )  =  ( I `  x
) )
3432, 33vtoclga 3017 . . . . . . . . 9  |-  ( ( R `  k )  e.  S  ->  (
I `  ( F `  ( R `  k
) ) )  =  ( I `  ( R `  k )
) )
3528, 34syl 16 . . . . . . . 8  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( F `  ( R `  k ) ) )  =  ( I `  ( R `  k ) ) )
3626, 35eqtrd 2468 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R `  k ) ) )
3736eqeq1d 2444 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) )  <->  ( I `  ( R `  k
) )  =  ( I `  ( R `
 0 ) ) ) )
3837biimprd 215 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( I `  ( R `  k ) )  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) )
3938expcom 425 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  S  ->  (
( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) )  -> 
( I `  ( R `  ( k  +  1 ) ) )  =  ( I `
 ( R ` 
0 ) ) ) ) )
4039a2d 24 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  S  -> 
( I `  ( R `  k )
)  =  ( I `
 ( R ` 
0 ) ) )  ->  ( A  e.  S  ->  ( I `  ( R `  (
k  +  1 ) ) )  =  ( I `  ( R `
 0 ) ) ) ) )
414, 8, 12, 16, 17, 40nn0ind 10366 . 2  |-  ( K  e.  NN0  ->  ( A  e.  S  ->  (
I `  ( R `  K ) )  =  ( I `  ( R `  0 )
) ) )
4241impcom 420 1  |-  ( ( A  e.  S  /\  K  e.  NN0 )  -> 
( I `  ( R `  K )
)  =  ( I `
 ( R ` 
0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814    X. cxp 4876    o. ccom 4882    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   0cc0 8990   1c1 8991    + caddc 8993   NN0cn0 10221   ZZcz 10282    seq cseq 11323
This theorem is referenced by:  eucalg  13078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324
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