Theorem algfx 12685

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 5789	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
4	1, 3	eqeltrid 2318	. . 3 |- ( A e. S -> N e. NN0 )
5	4	nn0zd 9643	. 2 |- ( A e. S -> N e. ZZ )
6		uzval 9800	. . . . . . 7 |- ( N e. ZZ -> ( ZZ>= ` N ) = { z e. ZZ | N <_ z } )
7	6	eleq2d 2301	. . . . . 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 5657	. . . . . . 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 5657	. . . . . . 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 5657	. . . . . . 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 5657	. . . . . . 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 2232	. . . . . . 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 2301	. . . . . . . . 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 9876	. . . . . . . . . . . . . . 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 9834	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
24		algcvga.2	. . . . . . . . . . . . . . 15 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
25		0zd 9534	. . . . . . . . . . . . . . 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 12679	. . . . . . . . . . . . . 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 12678	. . . . . . . . . . . . . . . 16 |- ( A e. S -> R : NN0 --> S )
32	31	ffvelcdmda 5790	. . . . . . . . . . . . . . 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 12684	. . . . . . . . . . . . . . 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 5657	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( C ` z ) = 0 <-> ( C ` ( R ` k ) ) = 0 ) )
38		fveq2 5648	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
39		id 19	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> z = ( R ` k ) )
40	38, 39	eqeq12d 2246	. . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 2264	. . . . . . . . . . . 12 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( R ` k ) )
46	45	eqeq1d 2240	. . . . . . . . . . 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 9636	. . . . 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 1398   e. wcel 2202   =/= wne 2403   {crab 2515   {csn 3673   class class class wbr 4093   X. cxp 4729   o. ccom 4735   -->wf 5329   ` cfv 5333  (class class class)co 6028   1stc1st 6310   0cc0 8075   1c1 8076   + caddc 8078   < clt 8257   <_ cle 8258   NN0cn0 9445   ZZcz 9522   ZZ>=cuz 9798   seqcseq 10753

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-z 9523  df-uz 9799  df-seqfrec 10754

This theorem is referenced by: (None)