Theorem algfx 13748

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	|- S e. _V
algcvga.2	|- F:S-->S
algcvga.3	|- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
algcvga.4	|- C:S-->NN0
algcvga.5	|- (z e. S -> ((C` (F` z)) =/= 0 -> (C` (F` z)) < (C` z)))
algcvga.6	|- N = (C` A)
algfx.7	|- (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

Proof of Theorem algfx

Step	Hyp	Ref	Expression
1		algcvga.4	. . . . 5 |- C:S-->NN0
2	1	ffvelrni 4788	. . . 4 |- (A e. S -> (C` A) e. NN0)
3		algcvga.6	. . . 4 |- N = (C` A)
4	2, 3	syl5eqel 1975	. . 3 |- (A e. S -> N e. NN0)
5		nn0z 7363	. . 3 |- (N e. NN0 -> N e. ZZ)
6	4, 5	syl 12	. 2 |- (A e. S -> N e. ZZ)
7		uzval 7588	. . . . . . 7 |- (N e. ZZ -> (ZZ>=` N) = {z e. ZZ | N <_ z})
8	7	eleq2d 1964	. . . . . 6 |- (N e. ZZ -> (K e. (ZZ>=` N) <-> K e. {z e. ZZ | N <_ z}))
9	8	pm5.32i 707	. . . . 5 |- ((N e. ZZ /\ K e. (ZZ>=` N)) <-> (N e. ZZ /\ K e. {z e. ZZ | N <_ z}))
10		fveq2 4681	. . . . . . . 8 |- (m = N -> (R` m) = (R` N))
11	10	eqeq1d 1892	. . . . . . 7 |- (m = N -> ((R` m) = (R` N) <-> (R` N) = (R` N)))
12	11	imbi2d 674	. . . . . 6 |- (m = N -> ((A e. S -> (R` m) = (R` N)) <-> (A e. S -> (R` N) = (R` N))))
13		fveq2 4681	. . . . . . . 8 |- (m = k -> (R` m) = (R` k))
14	13	eqeq1d 1892	. . . . . . 7 |- (m = k -> ((R` m) = (R` N) <-> (R` k) = (R` N)))
15	14	imbi2d 674	. . . . . 6 |- (m = k -> ((A e. S -> (R` m) = (R` N)) <-> (A e. S -> (R` k) = (R` N))))
16		fveq2 4681	. . . . . . . 8 |- (m = (k + 1) -> (R` m) = (R` (k + 1)))
17	16	eqeq1d 1892	. . . . . . 7 |- (m = (k + 1) -> ((R` m) = (R` N) <-> (R` (k + 1)) = (R` N)))
18	17	imbi2d 674	. . . . . 6 |- (m = (k + 1) -> ((A e. S -> (R` m) = (R` N)) <-> (A e. S -> (R` (k + 1)) = (R` N))))
19		fveq2 4681	. . . . . . . 8 |- (m = K -> (R` m) = (R` K))
20	19	eqeq1d 1892	. . . . . . 7 |- (m = K -> ((R` m) = (R` N) <-> (R` K) = (R` N)))
21	20	imbi2d 674	. . . . . 6 |- (m = K -> ((A e. S -> (R` m) = (R` N)) <-> (A e. S -> (R` K) = (R` N))))
22		eqidd 1885	. . . . . . 7 |- (A e. S -> (R` N) = (R` N))
23	22	a1i 8	. . . . . 6 |- (N e. ZZ -> (A e. S -> (R` N) = (R` N)))
24	7	eleq2d 1964	. . . . . . . . 9 |- (N e. ZZ -> (k e. (ZZ>=` N) <-> k e. {z e. ZZ | N <_ z}))
25	24	pm5.32i 707	. . . . . . . 8 |- ((N e. ZZ /\ k e. (ZZ>=` N)) <-> (N e. ZZ /\ k e. {z e. ZZ | N <_ z}))
26		simpl 346	. . . . . . . . . . . . . 14 |- ((A e. S /\ k e. (ZZ>=` N)) -> A e. S)
27		eluznn0 13661	. . . . . . . . . . . . . . 15 |- ((N e. NN0 /\ k e. (ZZ>=` N)) -> k e. NN0)
28	27, 4	sylan 497	. . . . . . . . . . . . . 14 |- ((A e. S /\ k e. (ZZ>=` N)) -> k e. NN0)
29		algcvga.1	. . . . . . . . . . . . . . 15 |- S e. _V
30		algcvga.2	. . . . . . . . . . . . . . 15 |- F:S-->S
31		algcvga.3	. . . . . . . . . . . . . . 15 |- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
32	29, 30, 31	algrp1 13742	. . . . . . . . . . . . . 14 |- ((A e. S /\ k e. NN0) -> (R` (k + 1)) = (F` (R` k)))
33	26, 28, 32	syl11anc 524	. . . . . . . . . . . . 13 |- ((A e. S /\ k e. (ZZ>=` N)) -> (R` (k + 1)) = (F` (R` k)))
34		algcvga.5	. . . . . . . . . . . . . . . 16 |- (z e. S -> ((C` (F` z)) =/= 0 -> (C` (F` z)) < (C` z)))
35	29, 30, 31, 1, 34, 3	algcvga 13747	. . . . . . . . . . . . . . 15 |- (A e. S -> (k e. (ZZ>=` N) -> (C` (R` k)) = 0))
36	35	imp 377	. . . . . . . . . . . . . 14 |- ((A e. S /\ k e. (ZZ>=` N)) -> (C` (R` k)) = 0)
37	26, 28	jca 310	. . . . . . . . . . . . . . 15 |- ((A e. S /\ k e. (ZZ>=` N)) -> (A e. S /\ k e. NN0))
38		ffvelrn 4787	. . . . . . . . . . . . . . . 16 |- ((R:NN0-->S /\ k e. NN0) -> (R` k) e. S)
39	29, 30, 31	algrf 13739	. . . . . . . . . . . . . . . 16 |- (A e. S -> R:NN0-->S)
40	38, 39	sylan 497	. . . . . . . . . . . . . . 15 |- ((A e. S /\ k e. NN0) -> (R` k) e. S)
41		fveq2 4681	. . . . . . . . . . . . . . . . . 18 |- (z = (R` k) -> (C` z) = (C` (R` k)))
42	41	eqeq1d 1892	. . . . . . . . . . . . . . . . 17 |- (z = (R` k) -> ((C` z) = 0 <-> (C` (R` k)) = 0))
43		fveq2 4681	. . . . . . . . . . . . . . . . . 18 |- (z = (R` k) -> (F` z) = (F` (R` k)))
44		id 73	. . . . . . . . . . . . . . . . . 18 |- (z = (R` k) -> z = (R` k))
45	43, 44	eqeq12d 1899	. . . . . . . . . . . . . . . . 17 |- (z = (R` k) -> ((F` z) = z <-> (F` (R` k)) = (R` k)))
46	42, 45	imbi12d 688	. . . . . . . . . . . . . . . 16 |- (z = (R` k) -> (((C` z) = 0 -> (F` z) = z) <-> ((C` (R` k)) = 0 -> (F` (R` k)) = (R` k))))
47		algfx.7	. . . . . . . . . . . . . . . . 17 |- (z e. S -> ((C` z) = 0 -> (F` z) = z))
48	47	rgen 2159	. . . . . . . . . . . . . . . 16 |- A.z e. S ((C` z) = 0 -> (F` z) = z)
49	46, 48	vtoclri 2360	. . . . . . . . . . . . . . 15 |- ((R` k) e. S -> ((C` (R` k)) = 0 -> (F` (R` k)) = (R` k)))
50	37, 40, 49	3syl 24	. . . . . . . . . . . . . 14 |- ((A e. S /\ k e. (ZZ>=` N)) -> ((C` (R` k)) = 0 -> (F` (R` k)) = (R` k)))
51	36, 50	mpd 29	. . . . . . . . . . . . 13 |- ((A e. S /\ k e. (ZZ>=` N)) -> (F` (R` k)) = (R` k))
52	33, 51	eqtrd 1925	. . . . . . . . . . . 12 |- ((A e. S /\ k e. (ZZ>=` N)) -> (R` (k + 1)) = (R` k))
53	52	eqeq1d 1892	. . . . . . . . . . 11 |- ((A e. S /\ k e. (ZZ>=` N)) -> ((R` (k + 1)) = (R` N) <-> (R` k) = (R` N)))
54	53	biimprd 171	. . . . . . . . . 10 |- ((A e. S /\ k e. (ZZ>=` N)) -> ((R` k) = (R` N) -> (R` (k + 1)) = (R` N)))
55	54	expcom 403	. . . . . . . . 9 |- (k e. (ZZ>=` N) -> (A e. S -> ((R` k) = (R` N) -> (R` (k + 1)) = (R` N))))
56	55	adantl 424	. . . . . . . 8 |- ((N e. ZZ /\ k e. (ZZ>=` N)) -> (A e. S -> ((R` k) = (R` N) -> (R` (k + 1)) = (R` N))))
57	25, 56	sylbir 218	. . . . . . 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 16	. . . . . 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	12, 15, 18, 21, 23, 58	uzind3 7419	. . . . 5 |- ((N e. ZZ /\ K e. {z e. ZZ | N <_ z}) -> (A e. S -> (R` K) = (R` N)))
60	9, 59	sylbi 216	. . . 4 |- ((N e. ZZ /\ K e. (ZZ>=` N)) -> (A e. S -> (R` K) = (R` N)))
61	60	ex 402	. . 3 |- (N e. ZZ -> (K e. (ZZ>=` N) -> (A e. S -> (R` K) = (R` N))))
62	61	com3r 39	. 2 |- (A e. S -> (N e. ZZ -> (K e. (ZZ>=` N) -> (R` K) = (R` N))))
63	6, 62	mpd 29	1 |- (A e. S -> (K e. (ZZ>=` N) -> (R` K) = (R` N)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  {crab 2108  _Vcvv 2292  {csn 3044   class class class wbr 3338   X. cxp 3984   |` cres 3988   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  0cc0 6386  1c1 6387   + caddc 6389   <_ cle 6448  NN0cn0 6450  ZZcz 6451   < clt 6653  ZZ>=cuz 7586   seq0 cseq0 7775

This theorem is referenced by:  mulgcdlem5 13760

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777

Copyright terms: Public domain