Theorem algfx 12589

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 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
algcvga.3	⊢ 𝐶:𝑆⟶ℕ0
algcvga.4	⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
algcvga.5	⊢ 𝑁 = (𝐶‘𝐴)
algfx.6	⊢ (𝑧 ∈ 𝑆 → ((𝐶‘𝑧) = 0 → (𝐹‘𝑧) = 𝑧))

Assertion

Ref	Expression
algfx	⊢ (𝐴 ∈ 𝑆 → (𝐾 ∈ (ℤ≥‘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⟶ℕ0
3	2	ffvelcdmi 5771	. . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0)
4	1, 3	eqeltrid 2316	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0)
5	4	nn0zd 9578	. 2 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℤ)
6		uzval 9735	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})
7	6	eleq2d 2299	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) ↔ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
8	7	pm5.32i 454	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
9		fveqeq2 5638	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝑁) = (𝑅‘𝑁)))
10	9	imbi2d 230	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁))))
11		fveqeq2 5638	. . . . . . 7 ⊢ (𝑚 = 𝑘 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝑘) = (𝑅‘𝑁)))
12	11	imbi2d 230	. . . . . 6 ⊢ (𝑚 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝑘) = (𝑅‘𝑁))))
13		fveqeq2 5638	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁)))
14	13	imbi2d 230	. . . . . 6 ⊢ (𝑚 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
15		fveqeq2 5638	. . . . . . 7 ⊢ (𝑚 = 𝐾 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝐾) = (𝑅‘𝑁)))
16	15	imbi2d 230	. . . . . 6 ⊢ (𝑚 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁))))
17		eqidd 2230	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁))
18	17	a1i 9	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁)))
19	6	eleq2d 2299	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (ℤ≥‘𝑁) ↔ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
20	19	pm5.32i 454	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
21		eluznn0 9806	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0)
22	4, 21	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0)
23		nn0uz 9769	. . . . . . . . . . . . . . 15 ⊢ ℕ0 = (ℤ≥‘0)
24		algcvga.2	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
25		0zd 9469	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
26		id 19	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
27		algcvga.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐹:𝑆⟶𝑆
28	27	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
29	23, 24, 25, 26, 28	algrp1 12583	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
30	22, 29	syldan 282	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
31	23, 24, 25, 26, 28	algrf 12582	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
32	31	ffvelcdmda 5772	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
33	22, 32	syldan 282	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝑘) ∈ 𝑆)
34		algcvga.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
35	27, 24, 2, 34, 1	algcvga 12588	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → (𝑘 ∈ (ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝑘)) = 0))
36	35	imp 124	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐶‘(𝑅‘𝑘)) = 0)
37		fveqeq2 5638	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘𝑧) = 0 ↔ (𝐶‘(𝑅‘𝑘)) = 0))
38		fveq2 5629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐹‘𝑧) = (𝐹‘(𝑅‘𝑘)))
39		id 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑅‘𝑘) → 𝑧 = (𝑅‘𝑘))
40	38, 39	eqeq12d 2244	. . . . . . . . . . . . . . . 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 ∈ 𝑆 → ((𝐶‘𝑧) = 0 → (𝐹‘𝑧) = 𝑧))
43	41, 42	vtoclga 2867	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘)))
44	33, 36, 43	sylc 62	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘))
45	30, 44	eqtrd 2262	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑘))
46	45	eqeq1d 2238	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑅‘(𝑘 + 1)) = (𝑅‘𝑁) ↔ (𝑅‘𝑘) = (𝑅‘𝑁)))
47	46	biimprd 158	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁)))
48	47	expcom 116	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
49	48	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
50	20, 49	sylbir 135	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
51	50	a2d 26	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → ((𝐴 ∈ 𝑆 → (𝑅‘𝑘) = (𝑅‘𝑁)) → (𝐴 ∈ 𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
52	10, 12, 14, 16, 18, 51	uzind3 9571	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁)))
53	8, 52	sylbi 121	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁)))
54	53	ex 115	. . 3 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁))))
55	54	com3r 79	. 2 ⊢ (𝐴 ∈ 𝑆 → (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) → (𝑅‘𝐾) = (𝑅‘𝑁))))
56	5, 55	mpd 13	1 ⊢ (𝐴 ∈ 𝑆 → (𝐾 ∈ (ℤ≥‘𝑁) → (𝑅‘𝐾) = (𝑅‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  {crab 2512  {csn 3666   class class class wbr 4083   × cxp 4717   ∘ ccom 4723  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  0cc0 8010  1c1 8011   + caddc 8013   < clt 8192   ≤ cle 8193  ℕ₀cn0 9380  ℤcz 9457  ℤ_≥cuz 9733  seqcseq 10681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-seqfrec 10682

This theorem is referenced by: (None)