Theorem alginv 12005

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 5504	. . . . 5 |- ( z = 0 -> ( I ` ( R ` z ) ) = ( I ` ( R ` 0 ) ) )
2	1	eqeq1d 2180	. . . 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 5504	. . . . 5 |- ( z = k -> ( I ` ( R ` z ) ) = ( I ` ( R ` k ) ) )
5	4	eqeq1d 2180	. . . 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 5504	. . . . 5 |- ( z = ( k + 1 ) -> ( I ` ( R ` z ) ) = ( I ` ( R ` ( k + 1 ) ) ) )
8	7	eqeq1d 2180	. . . 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 5504	. . . . 5 |- ( z = K -> ( I ` ( R ` z ) ) = ( I ` ( R ` K ) ) )
11	10	eqeq1d 2180	. . . 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 2172	. . 3 |- ( A e. S -> ( I ` ( R ` 0 ) ) = ( I ` ( R ` 0 ) ) )
14		nn0uz 9525	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
15		alginv.1	. . . . . . . . . 10 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
16		0zd 9228	. . . . . . . . . 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 12004	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
21	20	fveq2d 5503	. . . . . . . 8 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( F ` ( R ` k ) ) ) )
22	14, 15, 16, 17, 19	algrf 12003	. . . . . . . . . 10 |- ( A e. S -> R : NN0 --> S )
23	22	ffvelcdmda 5635	. . . . . . . . 9 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5504	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` ( F ` x ) ) = ( I ` ( F ` ( R ` k ) ) ) )
25		fveq2 5499	. . . . . . . . . . 11 |- ( x = ( R ` k ) -> ( I ` x ) = ( I ` ( R ` k ) ) )
26	24, 25	eqeq12d 2186	. . . . . . . . . 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 2797	. . . . . . . . 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 2204	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( I ` ( R ` ( k + 1 ) ) ) = ( I ` ( R ` k ) ) )
31	30	eqeq1d 2180	. . . . . 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 9330	. 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 1349   e. wcel 2142   {csn 3584   X. cxp 4610   o. ccom 4616   -->wf 5196   ` cfv 5200  (class class class)co 5857   1stc1st 6121   0cc0 7778   1c1 7779   + caddc 7781   NN0cn0 9139   seqcseq 10405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-addcom 7878  ax-addass 7880  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-0id 7886  ax-rnegex 7887  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-ltadd 7894

This theorem depends on definitions:  df-bi 116  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-inn 8883  df-n0 9140  df-z 9217  df-uz 9492  df-seqfrec 10406

This theorem is referenced by:  eucalg  12017