Theorem alginv 15288

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( z = 0 -> ( R ` z ) = ( R ` 0 ) )
2	1	fveq2d 6195	. . . . 5 |- ( z = 0 -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) )
3	2	eqeq1d 2624	. . . 4 |- ( z = 0 -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) )
4	3	imbi2d 330	. . 3 |- ( z = 0 -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) ) )
5		fveq2 6191	. . . . . 6 |- ( z = k -> ( R ` z ) = ( R ` k ) )
6	5	fveq2d 6195	. . . . 5 |- ( z = k -> ( I ` ( R ` z ) ) = ( I ` ( R ` k ) ) )
7	6	eqeq1d 2624	. . . 4 |- ( z = k -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
8	7	imbi2d 330	. . 3 |- ( z = k -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) ) )
9		fveq2 6191	. . . . . 6 |- ( z = ( k + 1 ) -> ( R ` z ) = ( R ` ( k + 1 ) ) )
10	9	fveq2d 6195	. . . . 5 |- ( z = ( k + 1 ) -> ( I ` ( R ` z ) ) = ( I ` ( R ` ( k + 1 ) ) ) )
11	10	eqeq1d 2624	. . . 4 |- ( z = ( k + 1 ) -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
12	11	imbi2d 330	. . 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 6191	. . . . . 6 |- ( z = K -> ( R ` z ) = ( R ` K ) )
14	13	fveq2d 6195	. . . . 5 |- ( z = K -> ( I ` ( R ` z ) ) = ( I ` ( R ` K ) ) )
15	14	eqeq1d 2624	. . . 4 |- ( z = K -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
16	15	imbi2d 330	. . 3 |- ( z = K -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) ) )
17		eqidd 2623	. . 3 |- ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) )
18		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
19		alginv.1	. . . . . . . . . 10 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
20		0zd 11389	. . . . . . . . . 10 |- ( A e. S -> 0 e. ZZ )
21		id 22	. . . . . . . . . 10 |- ( A e. S -> A e. S )
22		alginv.2	. . . . . . . . . . 11 |- F : S --> S
23	22	a1i 11	. . . . . . . . . 10 |- ( A e. S -> F : S --> S )
24	18, 19, 20, 21, 23	algrp1 15287	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
25	24	fveq2d 6195	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( F ` ( R ` k ) ) ) )
26	18, 19, 20, 21, 23	algrf 15286	. . . . . . . . . 10 |- ( A e. S -> R : NN0 --> S )
27	26	ffvelrnda 6359	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
28		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( R ` k ) -> ( F ` x ) = ( F ` ( R ` k ) ) )
29	28	fveq2d 6195	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` ( F ` x ) ) = ( I ` ( F ` ( R ` k ) ) ) )
30		fveq2 6191	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` x ) = ( I ` ( R ` k ) ) )
31	29, 30	eqeq12d 2637	. . . . . . . . . 10 |- ( x = ( R ` k ) -> ( ( I ` ( F ` x ) ) = ( I ` x ) <-> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) ) )
32		alginv.4	. . . . . . . . . 10 |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) )
33	31, 32	vtoclga 3272	. . . . . . . . 9 |- ( ( R ` k ) e. S -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
34	27, 33	syl 17	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
35	25, 34	eqtrd 2656	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` k ) ) )
36	35	eqeq1d 2624	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
37	36	biimprd 238	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
38	37	expcom 451	. . . 4 |- ( k e. NN0 -> ( A e. S -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
39	38	a2d 29	. . 3 |- ( k e. NN0 -> ( ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) -> ( A e. S -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
40	4, 8, 12, 16, 17, 39	nn0ind 11472	. 2 |- ( K e. NN0 -> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
41	40	impcom 446	1 |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  eucalg  15300