Theorem alginv 11521

Description: If I is an invariant of F, then 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	|- ( 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		2fveq3 5358	. . . . 5 |- ( z = 0 -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) )
2	1	eqeq1d 2108	. . . 4 |- ( z = 0 -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) )
3	2	imbi2d 229	. . 3 |- ( z = 0 -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) ) )
4		2fveq3 5358	. . . . 5 |- ( z = k -> ( I ` ( R ` z ) ) = ( I ` ( R ` k ) ) )
5	4	eqeq1d 2108	. . . 4 |- ( z = k -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
6	5	imbi2d 229	. . 3 |- ( z = k -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) ) )
7		2fveq3 5358	. . . . 5 |- ( z = ( k + 1 ) -> ( I ` ( R ` z ) ) = ( I ` ( R ` ( k + 1 ) ) ) )
8	7	eqeq1d 2108	. . . 4 |- ( z = ( k + 1 ) -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
9	8	imbi2d 229	. . 3 |- ( z = ( k + 1 ) -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
10		2fveq3 5358	. . . . 5 |- ( z = K -> ( I ` ( R ` z ) ) = ( I ` ( R ` K ) ) )
11	10	eqeq1d 2108	. . . 4 |- ( z = K -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
12	11	imbi2d 229	. . 3 |- ( z = K -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) ) )
13		eqidd 2101	. . 3 |- ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) )
14		nn0uz 9210	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15		alginv.1	. . . . . . . . . 10 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
16		0zd 8918	. . . . . . . . . 10 |- ( A e. S -> 0 e. ZZ )
17		id 19	. . . . . . . . . 10 |- ( A e. S -> A e. S )
18		alginv.2	. . . . . . . . . . 11 |- F : S --> S
19	18	a1i 9	. . . . . . . . . 10 |- ( A e. S -> F : S --> S )
20	14, 15, 16, 17, 19	algrp1 11520	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
21	20	fveq2d 5357	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( F ` ( R ` k ) ) ) )
22	14, 15, 16, 17, 19	algrf 11519	. . . . . . . . . 10 |- ( A e. S -> R : NN0 --> S )
23	22	ffvelrnda 5487	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5358	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` ( F ` x ) ) = ( I ` ( F ` ( R ` k ) ) ) )
25		fveq2 5353	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` x ) = ( I ` ( R ` k ) ) )
26	24, 25	eqeq12d 2114	. . . . . . . . . 10 |- ( x = ( R ` k ) -> ( ( I ` ( F ` x ) ) = ( I ` x ) <-> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) ) )
27		alginv.3	. . . . . . . . . 10 |- ( x e. S -> ( I ` ( F ` x ) ) = ( I ` x ) )
28	26, 27	vtoclga 2707	. . . . . . . . 9 |- ( ( R ` k ) e. S -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
29	23, 28	syl 14	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( F ` ( R ` k ) ) ) = ( I ` ( R ` k ) ) )
30	21, 29	eqtrd 2132	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` k ) ) )
31	30	eqeq1d 2108	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
32	31	biimprd 157	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
33	32	expcom 115	. . . 4 |- ( k e. NN0 -> ( A e. S -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) ) )
34	33	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 ) ) ) ) )
35	3, 6, 9, 12, 13, 34	nn0ind 9017	. 2 |- ( K e. NN0 -> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
36	35	impcom 124	1 |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   {csn 3474   X. cxp 4475   o. ccom 4481   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   0cc0 7500   1c1 7501   + caddc 7503   NN0cn0 8829   seqcseq 10059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060

This theorem is referenced by:  eucalg  11533