Theorem alginv 12049

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 5522	. . . . 5 |- ( z = 0 -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) )
2	1	eqeq1d 2186	. . . 4 |- ( z = 0 -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) )
3	2	imbi2d 230	. . 3 |- ( z = 0 -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) ) ) )
4		2fveq3 5522	. . . . 5 |- ( z = k -> ( I ` ( R ` z ) ) = ( I ` ( R ` k ) ) )
5	4	eqeq1d 2186	. . . 4 |- ( z = k -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
6	5	imbi2d 230	. . 3 |- ( z = k -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) ) )
7		2fveq3 5522	. . . . 5 |- ( z = ( k + 1 ) -> ( I ` ( R ` z ) ) = ( I ` ( R ` ( k + 1 ) ) ) )
8	7	eqeq1d 2186	. . . 4 |- ( z = ( k + 1 ) -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
9	8	imbi2d 230	. . 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 5522	. . . . 5 |- ( z = K -> ( I ` ( R ` z ) ) = ( I ` ( R ` K ) ) )
11	10	eqeq1d 2186	. . . 4 |- ( z = K -> ( ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
12	11	imbi2d 230	. . 3 |- ( z = K -> ( ( A e. S -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) ) <-> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) ) )
13		eqidd 2178	. . 3 |- ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) )
14		nn0uz 9564	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15		alginv.1	. . . . . . . . . 10 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
16		0zd 9267	. . . . . . . . . 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 12048	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
21	20	fveq2d 5521	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( F ` ( R ` k ) ) ) )
22	14, 15, 16, 17, 19	algrf 12047	. . . . . . . . . 10 |- ( A e. S -> R : NN0 --> S )
23	22	ffvelcdmda 5653	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5522	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` ( F ` x ) ) = ( I ` ( F ` ( R ` k ) ) ) )
25		fveq2 5517	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` x ) = ( I ` ( R ` k ) ) )
26	24, 25	eqeq12d 2192	. . . . . . . . . 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 2805	. . . . . . . . 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 2210	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` k ) ) )
31	30	eqeq1d 2186	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) <-> ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) ) )
32	31	biimprd 158	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( I ` ( R ` k ) ) = ( I ` ( R ` 0 ) ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` 0 ) ) ) )
33	32	expcom 116	. . . 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 9369	. 2 |- ( K e. NN0 -> ( A e. S -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) ) )
36	35	impcom 125	1 |- ( ( A e. S /\ K e. NN0 ) -> ( I ` ( R ` K ) ) = ( I ` ( R ` 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {csn 3594   X. cxp 4626   o. ccom 4632   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   0cc0 7813   1c1 7814   + caddc 7816   NN0cn0 9178   seqcseq 10447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-seqfrec 10448

This theorem is referenced by:  eucalg  12061