Theorem algfx 11526

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	ffvelrni 5486	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
4	1, 3	syl5eqel 2186	. . 3 |- ( A e. S -> N e. NN0 )
5	4	nn0zd 9023	. 2 |- ( A e. S -> N e. ZZ )
6		uzval 9178	. . . . . . 7 |- ( N e. ZZ -> ( ZZ>= ` N ) = { z e. ZZ | N <_ z } )
7	6	eleq2d 2169	. . . . . 6 |- ( N e. ZZ -> ( K e. ( ZZ>= ` N ) <-> K e. { z e. ZZ | N <_ z } ) )
8	7	pm5.32i 445	. . . . 5 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) )
9		fveqeq2 5362	. . . . . . 7 |- ( m = N -> ( ( R ` m ) = ( R ` N ) <-> ( R ` N ) = ( R ` N ) ) )
10	9	imbi2d 229	. . . . . 6 |- ( m = N -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` N ) = ( R ` N ) ) ) )
11		fveqeq2 5362	. . . . . . 7 |- ( m = k -> ( ( R ` m ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
12	11	imbi2d 229	. . . . . 6 |- ( m = k -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` k ) = ( R ` N ) ) ) )
13		fveqeq2 5362	. . . . . . 7 |- ( m = ( k + 1 ) -> ( ( R ` m ) = ( R ` N ) <-> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
14	13	imbi2d 229	. . . . . 6 |- ( m = ( k + 1 ) -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
15		fveqeq2 5362	. . . . . . 7 |- ( m = K -> ( ( R ` m ) = ( R ` N ) <-> ( R ` K ) = ( R ` N ) ) )
16	15	imbi2d 229	. . . . . 6 |- ( m = K -> ( ( A e. S -> ( R ` m ) = ( R ` N ) ) <-> ( A e. S -> ( R ` K ) = ( R ` N ) ) ) )
17		eqidd 2101	. . . . . . 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 2169	. . . . . . . . 9 |- ( N e. ZZ -> ( k e. ( ZZ>= ` N ) <-> k e. { z e. ZZ | N <_ z } ) )
20	19	pm5.32i 445	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) <-> ( N e. ZZ /\ k e. { z e. ZZ | N <_ z } ) )
21		eluznn0 9243	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
22	4, 21	sylan 279	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> k e. NN0 )
23		nn0uz 9210	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
24		algcvga.2	. . . . . . . . . . . . . . 15 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
25		0zd 8918	. . . . . . . . . . . . . . 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 11520	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
30	22, 29	syldan 278	. . . . . . . . . . . . 13 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
31	23, 24, 25, 26, 28	algrf 11519	. . . . . . . . . . . . . . . 16 |- ( A e. S -> R : NN0 --> S )
32	31	ffvelrnda 5487	. . . . . . . . . . . . . . 15 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
33	22, 32	syldan 278	. . . . . . . . . . . . . 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 11525	. . . . . . . . . . . . . . 15 |- ( A e. S -> ( k e. ( ZZ>= ` N ) -> ( C ` ( R ` k ) ) = 0 ) )
36	35	imp 123	. . . . . . . . . . . . . 14 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( C ` ( R ` k ) ) = 0 )
37		fveqeq2 5362	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( C ` z ) = 0 <-> ( C ` ( R ` k ) ) = 0 ) )
38		fveq2 5353	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
39		id 19	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> z = ( R ` k ) )
40	38, 39	eqeq12d 2114	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( F ` z ) = z <-> ( F ` ( R ` k ) ) = ( R ` k ) ) )
41	37, 40	imbi12d 233	. . . . . . . . . . . . . . 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 2707	. . . . . . . . . . . . . 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 2132	. . . . . . . . . . . 12 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( R ` k ) )
46	45	eqeq1d 2108	. . . . . . . . . . 11 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` ( k + 1 ) ) = ( R ` N ) <-> ( R ` k ) = ( R ` N ) ) )
47	46	biimprd 157	. . . . . . . . . 10 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) )
48	47	expcom 115	. . . . . . . . 9 |- ( k e. ( ZZ>= ` N ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
49	48	adantl 273	. . . . . . . 8 |- ( ( N e. ZZ /\ k e. ( ZZ>= ` N ) ) -> ( A e. S -> ( ( R ` k ) = ( R ` N ) -> ( R ` ( k + 1 ) ) = ( R ` N ) ) ) )
50	20, 49	sylbir 134	. . . . . . 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 9016	. . . . 5 |- ( ( N e. ZZ /\ K e. { z e. ZZ | N <_ z } ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
53	8, 52	sylbi 120	. . . 4 |- ( ( N e. ZZ /\ K e. ( ZZ>= ` N ) ) -> ( A e. S -> ( R ` K ) = ( R ` N ) ) )
54	53	ex 114	. . 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 103   = wceq 1299   e. wcel 1448   =/= wne 2267   {crab 2379   {csn 3474   class class class wbr 3875   X. cxp 4475   o. ccom 4481   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   0cc0 7500   1c1 7501   + caddc 7503   < clt 7672   <_ cle 7673   NN0cn0 8829   ZZcz 8906   ZZ>=cuz 9176   seqcseq 10059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060

This theorem is referenced by: (None)