Theorem algfx 12774

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 5816	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
4	1, 3	eqeltrid 2321	. . 3 |- ( A e. S -> N e. NN0 )
5	4	nn0zd 9716	. 2 |- ( A e. S -> N e. ZZ )
6		uzval 9873	. . . . . . 7 |- ( N e. ZZ -> ( ZZ>= ` N ) = { z e. ZZ | N <_ z } )
7	6	eleq2d 2304	. . . . . 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 5684	. . . . . . 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 5684	. . . . . . 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 5684	. . . . . . 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 5684	. . . . . . 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 2235	. . . . . . 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 2304	. . . . . . . . 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 9949	. . . . . . . . . . . . . . 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 9907	. . . . . . . . . . . . . . 15 |- NN0 = ( ZZ>= ` 0 )
24		algcvga.2	. . . . . . . . . . . . . . 15 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
25		0zd 9606	. . . . . . . . . . . . . . 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 12768	. . . . . . . . . . . . . 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 12767	. . . . . . . . . . . . . . . 16 |- ( A e. S -> R : NN0 --> S )
32	31	ffvelcdmda 5817	. . . . . . . . . . . . . . 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 12773	. . . . . . . . . . . . . . 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 5684	. . . . . . . . . . . . . . . 16 |- ( z = ( R ` k ) -> ( ( C ` z ) = 0 <-> ( C ` ( R ` k ) ) = 0 ) )
38		fveq2 5675	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
39		id 19	. . . . . . . . . . . . . . . . 17 |- ( z = ( R ` k ) -> z = ( R ` k ) )
40	38, 39	eqeq12d 2249	. . . . . . . . . . . . . . . 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 2883	. . . . . . . . . . . . . 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 2267	. . . . . . . . . . . 12 |- ( ( A e. S /\ k e. ( ZZ>= ` N ) ) -> ( R ` ( k + 1 ) ) = ( R ` k ) )
46	45	eqeq1d 2243	. . . . . . . . . . 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 9709	. . . . 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 2205   =/= wne 2414   {crab 2526   {csn 3694   class class class wbr 4114   X. cxp 4752   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   seqcseq 10833

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834

This theorem is referenced by: (None)