Theorem algfx 12220

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 } ) )
algcvga.3	|- C : S --> NN0
algcvga.4	|- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
algcvga.5	|- N = ( C ` A )
algfx.6	|- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )

Assertion

Ref	Expression
algfx	|- ( 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 hint:   A( z)

Proof of Theorem algfx

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	ffvelcdmi 5696	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
4	1, 3	eqeltrid 2283	. . 3 |- ( A e. S -> N e. NN0 )
5	4	nn0zd 9446	. 2 |- ( A e. S -> N e. ZZ )
6		uzval 9603	. . . . . . 7 |- ( N e. ZZ -> ( ZZ>= ` N ) = { z e. ZZ | N <_ z } )
7	6	eleq2d 2266	. . . . . 6 |- ( N e. ZZ -> ( K e. ( ZZ>= ` N ) <-> K e. { z e. ZZ | N <_ z } ) )
8	7	pm5.32i 454	. . . . 5 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) )
9		fveqeq2 5567	. . . . . . 7 |- ( m = N -> ( ( R ` m ) = ( R ` N ) <-> ( R ` N ) = ( R ` N ) ) )
10	9	imbi2d 230	. . . . . 6 |- ( m = N -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` N ) = ( R ` N ) ) ) )
11		fveqeq2 5567	. . . . . . 7 |- ( m = k -> ( ( R ` m ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
12	11	imbi2d 230	. . . . . 6 |- ( m = k -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` k ) = ( R ` N ) ) ) )
13		fveqeq2 5567	. . . . . . 7 |- ( m = ( k + 1 ) -> ( ( R ` m ) = ( R ` N ) <-> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
14	13	imbi2d 230	. . . . . 6 |- ( m = ( k + 1 ) -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
15		fveqeq2 5567	. . . . . . 7 |- ( m = K -> ( ( R ` m ) = ( R ` N ) <-> ( R ` K ) = ( R ` N ) ) )
16	15	imbi2d 230	. . . . . 6 |- ( m = K -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` K ) = ( R ` N ) ) ) )
17		eqidd 2197	. . . . . . 7 |- ( A e. S -> ( R ` N ) = ( R ` N ) )
18	17	a1i 9	. . . . . 6 |- ( N e. ZZ -> ( A e. S -> ( R ` N ) = ( R ` N ) ) )
19	6	eleq2d 2266	. . . . . . . . 9 |- ( N e. ZZ -> ( k e. ( ZZ>= ` N ) <-> k e. { z e. ZZ | N <_ z } ) )
20	19	pm5.32i 454	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) )
21		eluznn0 9673	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
22	4, 21	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
23		nn0uz 9636	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
24		algcvga.2	. . . . . . . . . . . . . . 15 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
25		0zd 9338	. . . . . . . . . . . . . . 15 |- ( A e. S -> 0 e. ZZ )
26		id 19	. . . . . . . . . . . . . . 15 |- ( A e. S -> A e. S )
27		algcvga.1	. . . . . . . . . . . . . . . 16 |- F : S --> S
28	27	a1i 9	. . . . . . . . . . . . . . 15 |- ( A e. S -> F : S --> S )
29	23, 24, 25, 26, 28	algrp1 12214	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
30	22, 29	syldan 282	. . . . . . . . . . . . 13 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
31	23, 24, 25, 26, 28	algrf 12213	. . . . . . . . . . . . . . . 16 |- ( A e. S -> R : NN0 --> S )
32	31	ffvelcdmda 5697	. . . . . . . . . . . . . . 15 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
33	22, 32	syldan 282	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` k ) e. S )
34		algcvga.4	. . . . . . . . . . . . . . . 16 |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
35	27, 24, 2, 34, 1	algcvga 12219	. . . . . . . . . . . . . . 15 |- ( A e. S -> ( k e. ( ZZ>= ` N ) -> ( C ` ( R ` k ) ) = 0 ) )
36	35	imp 124	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( C ` ( R ` k ) ) = 0 )
37		fveqeq2 5567	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( C ` z ) = 0 <-> ( C ` ( R ` k ) ) = 0 ) )
38		fveq2 5558	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
39		id 19	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> z = ( R ` k ) )
40	38, 39	eqeq12d 2211	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( F ` z ) = z <-> ( F ` ( R ` k ) ) = ( R ` k ) ) )
41	37, 40	imbi12d 234	. . . . . . . . . . . . . . 15 |- ( z = ( R ` k ) -> ( ( ( C ` z ) = 0 -> ( F ` z ) = z ) <-> ( ( C ` ( R ` k ) ) = 0 -> ( F ` ( R ` k ) ) = ( R ` k ) ) ) )
42		algfx.6	. . . . . . . . . . . . . . 15 |- ( z e. S -> ( ( C ` z ) = 0 -> ( F ` z ) = z ) )
43	41, 42	vtoclga 2830	. . . . . . . . . . . . . 14 |- ( ( R ` k ) e. S -> ( ( C ` ( R ` k ) ) = 0 -> ( F ` ( R ` k ) ) = ( R ` k ) ) )
44	33, 36, 43	sylc 62	. . . . . . . . . . . . 13 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( F ` ( R ` k ) ) = ( R ` k ) )
45	30, 44	eqtrd 2229	. . . . . . . . . . . 12 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( R ` k ) )
46	45	eqeq1d 2205	. . . . . . . . . . 11 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` ( k + 1 ) ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
47	46	biimprd 158	. . . . . . . . . 10 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
48	47	expcom 116	. . . . . . . . 9 |- ( k e. ( ZZ>= ` N ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
49	48	adantl 277	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
50	20, 49	sylbir 135	. . . . . . 7 |- ( ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
51	50	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 ) ) ) )
52	10, 12, 14, 16, 18, 51	uzind3 9439	. . . . 5 |- ( ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
53	8, 52	sylbi 121	. . . 4 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
54	53	ex 115	. . 3 |- ( N e. ZZ -> ( K e. ( ZZ>= ` N ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) ) )
55	54	com3r 79	. 2 |- ( A e. S -> ( N e. ZZ -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) ) )
56	5, 55	mpd 13	1 |- ( A e. S -> ( K e. ( ZZ>= ` N ) -> ( R ` K ) = ( R ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   {crab 2479   {csn 3622   class class class wbr 4033   X. cxp 4661   o. ccom 4667   -->wf 5254   ` cfv 5258  (class class class)co 5922   1stc1st 6196   0cc0 7879   1c1 7880   + caddc 7882   < clt 8061   <_ cle 8062   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   seqcseq 10539

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540

This theorem is referenced by: (None)