Theorem algfx 15131

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	ffvelrni 6266	. . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0)
4	1, 3	syl5eqel 2692	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0)
5	4	nn0zd 11356	. 2 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℤ)
6		uzval 11565	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})
7	6	eleq2d 2673	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) ↔ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
8	7	pm5.32i 667	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
9		fveq2 6103	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑅‘𝑚) = (𝑅‘𝑁))
10	9	eqeq1d 2612	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝑁) = (𝑅‘𝑁)))
11	10	imbi2d 329	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁))))
12		fveq2 6103	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝑅‘𝑚) = (𝑅‘𝑘))
13	12	eqeq1d 2612	. . . . . . 7 ⊢ (𝑚 = 𝑘 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝑘) = (𝑅‘𝑁)))
14	13	imbi2d 329	. . . . . 6 ⊢ (𝑚 = 𝑘 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝑘) = (𝑅‘𝑁))))
15		fveq2 6103	. . . . . . . 8 ⊢ (𝑚 = (𝑘 + 1) → (𝑅‘𝑚) = (𝑅‘(𝑘 + 1)))
16	15	eqeq1d 2612	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁)))
17	16	imbi2d 329	. . . . . 6 ⊢ (𝑚 = (𝑘 + 1) → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
18		fveq2 6103	. . . . . . . 8 ⊢ (𝑚 = 𝐾 → (𝑅‘𝑚) = (𝑅‘𝐾))
19	18	eqeq1d 2612	. . . . . . 7 ⊢ (𝑚 = 𝐾 → ((𝑅‘𝑚) = (𝑅‘𝑁) ↔ (𝑅‘𝐾) = (𝑅‘𝑁)))
20	19	imbi2d 329	. . . . . 6 ⊢ (𝑚 = 𝐾 → ((𝐴 ∈ 𝑆 → (𝑅‘𝑚) = (𝑅‘𝑁)) ↔ (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁))))
21		eqidd 2611	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁))
22	21	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝐴 ∈ 𝑆 → (𝑅‘𝑁) = (𝑅‘𝑁)))
23	6	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (ℤ≥‘𝑁) ↔ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
24	23	pm5.32i 667	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))
25		eluznn0 11633	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0)
26	4, 25	sylan 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ ℕ0)
27		nn0uz 11598	. . . . . . . . . . . . . . 15 ⊢ ℕ0 = (ℤ≥‘0)
28		algcvga.2	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
29		0zd 11266	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
30		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
31		algcvga.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐹:𝑆⟶𝑆
32	31	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
33	27, 28, 29, 30, 32	algrp1 15125	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
34	26, 33	syldan 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
35	27, 28, 29, 30, 32	algrf 15124	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
36	35	ffvelrnda 6267	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
37	26, 36	syldan 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘𝑘) ∈ 𝑆)
38		algcvga.4	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
39	31, 28, 2, 38, 1	algcvga 15130	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑆 → (𝑘 ∈ (ℤ≥‘𝑁) → (𝐶‘(𝑅‘𝑘)) = 0))
40	39	imp 444	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐶‘(𝑅‘𝑘)) = 0)
41		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘)))
42	41	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘𝑧) = 0 ↔ (𝐶‘(𝑅‘𝑘)) = 0))
43		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐹‘𝑧) = (𝐹‘(𝑅‘𝑘)))
44		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑅‘𝑘) → 𝑧 = (𝑅‘𝑘))
45	43, 44	eqeq12d 2625	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐹‘𝑧) = 𝑧 ↔ (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘)))
46	42, 45	imbi12d 333	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅‘𝑘) → (((𝐶‘𝑧) = 0 → (𝐹‘𝑧) = 𝑧) ↔ ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘))))
47		algfx.6	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘𝑧) = 0 → (𝐹‘𝑧) = 𝑧))
48	46, 47	vtoclga 3245	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝑅‘𝑘)) = 0 → (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘)))
49	37, 40, 48	sylc 63	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘(𝑅‘𝑘)) = (𝑅‘𝑘))
50	34, 49	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑘))
51	50	eqeq1d 2612	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑅‘(𝑘 + 1)) = (𝑅‘𝑁) ↔ (𝑅‘𝑘) = (𝑅‘𝑁)))
52	51	biimprd 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁)))
53	52	expcom 450	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
54	53	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
55	24, 54	sylbir 224	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝐴 ∈ 𝑆 → ((𝑅‘𝑘) = (𝑅‘𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
56	55	a2d 29	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → ((𝐴 ∈ 𝑆 → (𝑅‘𝑘) = (𝑅‘𝑁)) → (𝐴 ∈ 𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅‘𝑁))))
57	11, 14, 17, 20, 22, 56	uzind3 11347	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁)))
58	8, 57	sylbi 206	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁)))
59	58	ex 449	. . 3 ⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) → (𝐴 ∈ 𝑆 → (𝑅‘𝐾) = (𝑅‘𝑁))))
60	59	com3r 85	. 2 ⊢ (𝐴 ∈ 𝑆 → (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ≥‘𝑁) → (𝑅‘𝐾) = (𝑅‘𝑁))))
61	5, 60	mpd 15	1 ⊢ (𝐴 ∈ 𝑆 → (𝐾 ∈ (ℤ≥‘𝑁) → (𝑅‘𝐾) = (𝑅‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  {csn 4125   class class class wbr 4583   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  seqcseq 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664

This theorem is referenced by: (None)