Theorem alginv 13743

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
algrp1.1	|- S e. _V
algrp1.2	|- F:S-->S
algrp1.3	|- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
alginv.4	|- I Fn S
alginv.5	|- (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

Proof of Theorem alginv

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . 6 |- (m = 0 -> (R` m) = (R` 0))
2	1	fveq2d 4685	. . . . 5 |- (m = 0 -> (I` (R` m)) = (I` (R` 0)))
3	2	eqeq1d 1892	. . . 4 |- (m = 0 -> ((I` (R` m)) = (I` (R` 0)) <-> (I` (R` 0)) = (I` (R` 0))))
4	3	imbi2d 674	. . 3 |- (m = 0 -> ((A e. S -> (I` (R` m)) = (I` (R` 0))) <-> (A e. S -> (I` (R` 0)) = (I` (R` 0)))))
5		fveq2 4681	. . . . . 6 |- (m = k -> (R` m) = (R` k))
6	5	fveq2d 4685	. . . . 5 |- (m = k -> (I` (R` m)) = (I` (R` k)))
7	6	eqeq1d 1892	. . . 4 |- (m = k -> ((I` (R` m)) = (I` (R` 0)) <-> (I` (R` k)) = (I` (R` 0))))
8	7	imbi2d 674	. . 3 |- (m = k -> ((A e. S -> (I` (R` m)) = (I` (R` 0))) <-> (A e. S -> (I` (R` k)) = (I` (R` 0)))))
9		fveq2 4681	. . . . . 6 |- (m = (k + 1) -> (R` m) = (R` (k + 1)))
10	9	fveq2d 4685	. . . . 5 |- (m = (k + 1) -> (I` (R` m)) = (I` (R` (k + 1))))
11	10	eqeq1d 1892	. . . 4 |- (m = (k + 1) -> ((I` (R` m)) = (I` (R` 0)) <-> (I` (R` (k + 1))) = (I` (R` 0))))
12	11	imbi2d 674	. . 3 |- (m = (k + 1) -> ((A e. S -> (I` (R` m)) = (I` (R` 0))) <-> (A e. S -> (I` (R` (k + 1))) = (I` (R` 0)))))
13		fveq2 4681	. . . . . 6 |- (m = K -> (R` m) = (R` K))
14	13	fveq2d 4685	. . . . 5 |- (m = K -> (I` (R` m)) = (I` (R` K)))
15	14	eqeq1d 1892	. . . 4 |- (m = K -> ((I` (R` m)) = (I` (R` 0)) <-> (I` (R` K)) = (I` (R` 0))))
16	15	imbi2d 674	. . 3 |- (m = K -> ((A e. S -> (I` (R` m)) = (I` (R` 0))) <-> (A e. S -> (I` (R` K)) = (I` (R` 0)))))
17		eqidd 1885	. . 3 |- (A e. S -> (I` (R` 0)) = (I` (R` 0)))
18		algrp1.1	. . . . . . . . . 10 |- S e. _V
19		algrp1.2	. . . . . . . . . 10 |- F:S-->S
20		algrp1.3	. . . . . . . . . 10 |- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
21	18, 19, 20	algrp1 13742	. . . . . . . . 9 |- ((A e. S /\ k e. NN0) -> (R` (k + 1)) = (F` (R` k)))
22	21	fveq2d 4685	. . . . . . . 8 |- ((A e. S /\ k e. NN0) -> (I` (R` (k + 1))) = (I` (F` (R` k))))
23		ffvelrn 4787	. . . . . . . . . 10 |- ((R:NN0-->S /\ k e. NN0) -> (R` k) e. S)
24	18, 19, 20	algrf 13739	. . . . . . . . . 10 |- (A e. S -> R:NN0-->S)
25	23, 24	sylan 497	. . . . . . . . 9 |- ((A e. S /\ k e. NN0) -> (R` k) e. S)
26		fveq2 4681	. . . . . . . . . . . 12 |- (x = (R` k) -> (F` x) = (F` (R` k)))
27	26	fveq2d 4685	. . . . . . . . . . 11 |- (x = (R` k) -> (I` (F` x)) = (I` (F` (R` k))))
28		fveq2 4681	. . . . . . . . . . 11 |- (x = (R` k) -> (I` x) = (I` (R` k)))
29	27, 28	eqeq12d 1899	. . . . . . . . . 10 |- (x = (R` k) -> ((I` (F` x)) = (I` x) <-> (I` (F` (R` k))) = (I` (R` k))))
30		alginv.5	. . . . . . . . . . 11 |- (x e. S -> (I` (F` x)) = (I` x))
31	30	rgen 2159	. . . . . . . . . 10 |- A.x e. S (I` (F` x)) = (I` x)
32	29, 31	vtoclri 2360	. . . . . . . . 9 |- ((R` k) e. S -> (I` (F` (R` k))) = (I` (R` k)))
33	25, 32	syl 12	. . . . . . . 8 |- ((A e. S /\ k e. NN0) -> (I` (F` (R` k))) = (I` (R` k)))
34	22, 33	eqtrd 1925	. . . . . . 7 |- ((A e. S /\ k e. NN0) -> (I` (R` (k + 1))) = (I` (R` k)))
35	34	eqeq1d 1892	. . . . . 6 |- ((A e. S /\ k e. NN0) -> ((I` (R` (k + 1))) = (I` (R` 0)) <-> (I` (R` k)) = (I` (R` 0))))
36	35	biimprd 171	. . . . 5 |- ((A e. S /\ k e. NN0) -> ((I` (R` k)) = (I` (R` 0)) -> (I` (R` (k + 1))) = (I` (R` 0))))
37	36	expcom 403	. . . 4 |- (k e. NN0 -> (A e. S -> ((I` (R` k)) = (I` (R` 0)) -> (I` (R` (k + 1))) = (I` (R` 0)))))
38	37	a2d 16	. . 3 |- (k e. NN0 -> ((A e. S -> (I` (R` k)) = (I` (R` 0))) -> (A e. S -> (I` (R` (k + 1))) = (I` (R` 0)))))
39	4, 8, 12, 16, 17, 38	nn0ind 7424	. 2 |- (K e. NN0 -> (A e. S -> (I` (R` K)) = (I` (R` 0))))
40	39	impcom 378	1 |- ((A e. S /\ K e. NN0) -> (I` (R` K)) = (I` (R` 0)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044   X. cxp 3984   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  0cc0 6386  1c1 6387   + caddc 6389  NN0cn0 6450   seq0 cseq0 7775

This theorem is referenced by:  eucalg 13755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777

Copyright terms: Public domain