Theorem ialgfx 10434

Description: If F reaches a fixed point when the countdown function C reaches 0, F remains fixed after N steps. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
algcvga.1	|- F : S --> S
algcvga.2	|- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )
algcvga.3	|- C : S --> NN0
algcvga.4	|- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
algcvga.5	|- N = ( C ` A )
ialgcvga.s	|- S e. V
algfx.6	|- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )

Assertion

Ref	Expression
ialgfx	|- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )

Distinct variable groups:   z, C   z, F   z, R   z, S   z, K   z, N

Allowed substitution hints:   A( z)   V( z)

Proof of Theorem ialgfx

Dummy variables k m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		algcvga.5	. . . 4 |- N = ( C ` A )
2		algcvga.3	. . . . 5 |- C : S --> NN0
3	2	ffvelrni 5322	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
4	1, 3	syl5eqel 2165	. . 3 |- ( A e. S -> N e. NN0 )
5	4	nn0zd 8467	. 2 |- ( A e. S -> N e. ZZ )
6		uzval 8621	. . . . . . 7 |- ( N e. ZZ -> ( ZZ>= ` N ) = { z e. ZZ | N <_ z } )
7	6	eleq2d 2148	. . . . . 6 |- ( N e. ZZ -> ( K e. ( ZZ>= ` N ) <-> K e. { z e. ZZ | N <_ z } ) )
8	7	pm5.32i 441	. . . . 5 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) )
9		fveq2 5198	. . . . . . . 8 |- ( m = N -> ( R ` m ) = ( R ` N ) )
10	9	eqeq1d 2089	. . . . . . 7 |- ( m = N -> ( ( R ` m ) = ( R ` N ) <-> ( R ` N ) = ( R ` N ) ) )
11	10	imbi2d 228	. . . . . 6 |- ( m = N -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` N ) = ( R ` N ) ) ) )
12		fveq2 5198	. . . . . . . 8 |- ( m = k -> ( R ` m ) = ( R ` k ) )
13	12	eqeq1d 2089	. . . . . . 7 |- ( m = k -> ( ( R ` m ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
14	13	imbi2d 228	. . . . . 6 |- ( m = k -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` k ) = ( R ` N ) ) ) )
15		fveq2 5198	. . . . . . . 8 |- ( m = ( k + 1 ) -> ( R ` m ) = ( R ` ( k + 1 ) ) )
16	15	eqeq1d 2089	. . . . . . 7 |- ( m = ( k + 1 ) -> ( ( R ` m ) = ( R ` N ) <-> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
17	16	imbi2d 228	. . . . . 6 |- ( m = ( k + 1 ) -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
18		fveq2 5198	. . . . . . . 8 |- ( m = K -> ( R ` m ) = ( R ` K ) )
19	18	eqeq1d 2089	. . . . . . 7 |- ( m = K -> ( ( R ` m ) = ( R ` N ) <-> ( R ` K ) = ( R ` N ) ) )
20	19	imbi2d 228	. . . . . 6 |- ( m = K -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` K ) = ( R ` N ) ) ) )
21		eqidd 2082	. . . . . . 7 |- ( A e. S -> ( R ` N ) = ( R ` N ) )
22	21	a1i 9	. . . . . 6 |- ( N e. ZZ -> ( A e. S -> ( R ` N ) = ( R ` N ) ) )
23	6	eleq2d 2148	. . . . . . . . 9 |- ( N e. ZZ -> ( k e. ( ZZ>= ` N ) <-> k e. { z e. ZZ | N <_ z } ) )
24	23	pm5.32i 441	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) )
25		eluznn0 8686	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
26	4, 25	sylan 277	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
27		nn0uz 8653	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
28		algcvga.2	. . . . . . . . . . . . . . 15 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )
29		0zd 8363	. . . . . . . . . . . . . . 15 |- ( A e. S -> 0 e. ZZ )
30		id 19	. . . . . . . . . . . . . . 15 |- ( A e. S -> A e. S )
31		algcvga.1	. . . . . . . . . . . . . . . 16 |- F : S --> S
32	31	a1i 9	. . . . . . . . . . . . . . 15 |- ( A e. S -> F : S --> S )
33		ialgcvga.s	. . . . . . . . . . . . . . . 16 |- S e. V
34	33	a1i 9	. . . . . . . . . . . . . . 15 |- ( A e. S -> S e. V )
35	27, 28, 29, 30, 32, 34	ialgrp1 10428	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
36	26, 35	syldan 276	. . . . . . . . . . . . 13 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
37	27, 28, 29, 30, 32, 34	ialgrf 10427	. . . . . . . . . . . . . . . 16 |- ( A e. S -> R : NN0 --> S )
38	37	ffvelrnda 5323	. . . . . . . . . . . . . . 15 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
39	26, 38	syldan 276	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` k ) e. S )
40		algcvga.4	. . . . . . . . . . . . . . . 16 |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
41	31, 28, 2, 40, 1, 33	ialgcvga 10433	. . . . . . . . . . . . . . 15 |- ( A e. S -> ( k e. ( ZZ>= ` N ) -> ( C ` ( R ` k ) ) = 0 ) )
42	41	imp 122	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( C ` ( R ` k ) ) = 0 )
43		fveq2 5198	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
44	43	eqeq1d 2089	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( C ` z ) = 0 <-> ( C ` ( R ` k ) ) = 0 ) )
45		fveq2 5198	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
46		id 19	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> z = ( R ` k ) )
47	45, 46	eqeq12d 2095	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( F ` z ) = z <-> ( F ` ( R ` k ) ) = ( R ` k ) ) )
48	44, 47	imbi12d 232	. . . . . . . . . . . . . . 15 |- ( z = ( R ` k ) -> ( ( ( C ` z ) = 0 -> ( F ` z ) = z ) <-> ( ( C ` ( R ` k ) ) = 0 -> ( F ` ( R ` k ) ) = ( R ` k ) ) ) )
49		algfx.6	. . . . . . . . . . . . . . 15 |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )
50	48, 49	vtoclga 2664	. . . . . . . . . . . . . 14 |- ( ( R ` k ) e. S -> ( ( C ` ( R ` k ) ) = 0 -> ( F ` ( R ` k ) ) = ( R ` k ) ) )
51	39, 42, 50	sylc 61	. . . . . . . . . . . . 13 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( F ` ( R ` k ) ) = ( R ` k ) )
52	36, 51	eqtrd 2113	. . . . . . . . . . . 12 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( R ` k ) )
53	52	eqeq1d 2089	. . . . . . . . . . 11 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` ( k + 1 ) ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
54	53	biimprd 156	. . . . . . . . . 10 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
55	54	expcom 114	. . . . . . . . 9 |- ( k e. ( ZZ>= ` N ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
56	55	adantl 271	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
57	24, 56	sylbir 133	. . . . . . 7 |- ( ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
58	57	a2d 26	. . . . . 6 |- ( ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) -> ( ( A e. S -> ( R ` k ) = ( R ` N ) ) -> ( A e. S -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
59	11, 14, 17, 20, 22, 58	uzind3 8460	. . . . 5 |- ( ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
60	8, 59	sylbi 119	. . . 4 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
61	60	ex 113	. . 3 |- ( N e. ZZ -> ( K e. ( ZZ>= ` N ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) ) )
62	61	com3r 78	. 2 |- ( A e. S -> ( N e. ZZ -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) ) )
63	5, 62	mpd 13	1 |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   =/= wne 2245   {crab 2352   {csn 3398   class class class wbr 3785   X. cxp 4361   o. ccom 4367   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   0cc0 6981   1c1 6982   + caddc 6984   < clt 7153   <_ cle 7154   NN0cn0 8288   ZZcz 8351   ZZ>=cuz 8619   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-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-dc 776  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: (None)