Theorem ialginv 10429

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.

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . . . 6 |- ( z = 0 -> ( R ` z ) = ( R ` 0 ) )
2	1	fveq2d 5202	. . . . 5 |- ( z = 0 -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) )
3	2	eqeq1d 2089	. . . 4 |- ( z = 0 -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) )
4	3	imbi2d 228	. . 3 |- ( z = 0 -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) ) )
5		fveq2 5198	. . . . . 6 |- ( z = k -> ( R ` z ) = ( R ` k ) )
6	5	fveq2d 5202	. . . . 5 |- ( z = k -> ( I ` ( R ` z ) ) = ( I ` ( R ` k ) ) )
7	6	eqeq1d 2089	. . . 4 |- ( z = k -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
8	7	imbi2d 228	. . 3 |- ( z = k -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) ) )
9		fveq2 5198	. . . . . 6 |- ( z = ( k + 1 ) -> ( R ` z ) = ( R ` ( k + 1 ) ) )
10	9	fveq2d 5202	. . . . 5 |- ( z = ( k + 1 ) -> ( I ` ( R ` z ) ) = ( I ` ( R ` ( k + 1 ) ) ) )
11	10	eqeq1d 2089	. . . 4 |- ( z = ( k + 1 ) -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
12	11	imbi2d 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 5198	. . . . . 6 |- ( z = K -> ( R ` z ) = ( R ` K ) )
14	13	fveq2d 5202	. . . . 5 |- ( z = K -> ( I ` ( R ` z ) ) = ( I ` ( R ` K ) ) )
15	14	eqeq1d 2089	. . . 4 |- ( z = K -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
16	15	imbi2d 228	. . 3 |- ( z = K -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) ) )
17		eqidd 2082	. . 3 |- ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) )
18		nn0uz 8653	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
19		alginv.1	. . . . . . . . . 10 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )
20		0zd 8363	. . . . . . . . . 10 |- ( A e. S -> 0 e. ZZ )
21		id 19	. . . . . . . . . 10 |- ( A e. S -> A e. S )
22		alginv.2	. . . . . . . . . . 11 |- F : S --> S
23	22	a1i 9	. . . . . . . . . 10 |- ( A e. S -> F : S --> S )
24		alginv.s	. . . . . . . . . . 11 |- S e. V
25	24	a1i 9	. . . . . . . . . 10 |- ( A e. S -> S e. V )
26	18, 19, 20, 21, 23, 25	ialgrp1 10428	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
27	26	fveq2d 5202	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( F ` ( R ` k ) ) ) )
28	18, 19, 20, 21, 23, 25	ialgrf 10427	. . . . . . . . . 10 |- ( A e. S -> R : NN0 --> S )
29	28	ffvelrnda 5323	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
30		fveq2 5198	. . . . . . . . . . . 12 |- ( x = ( R ` k ) -> ( F ` x ) = ( F ` ( R ` k ) ) )
31	30	fveq2d 5202	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` ( F ` x ) ) = ( I ` ( F ` ( R ` k ) ) ) )
32		fveq2 5198	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` x ) = ( I ` ( R ` k ) ) )
33	31, 32	eqeq12d 2095	. . . . . . . . . 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 ) )
35	33, 34	vtoclga 2664	. . . . . . . . 9 |- ( ( R ` k ) e. S -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
36	29, 35	syl 14	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
37	27, 36	eqtrd 2113	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` k ) ) )
38	37	eqeq1d 2089	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
39	38	biimprd 156	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
40	39	expcom 114	. . . 4 |- ( k e. NN0 -> ( A e. S -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
41	40	a2d 26	. . 3 |- ( k e. NN0 -> ( ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) -> ( A e. S -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
42	4, 8, 12, 16, 17, 41	nn0ind 8461	. 2 |- ( K e. NN0 -> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
43	42	impcom 123	1 |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {csn 3398   X. cxp 4361   o. ccom 4367   Fn wfn 4917   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   0cc0 6981   1c1 6982   + caddc 6984   NN0cn0 8288   seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  eucialg  10441