ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  algfx GIF version

Theorem algfx 11461
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:𝑆𝑆
algcvga.2 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
algcvga.3 𝐶:𝑆⟶ℕ0
algcvga.4 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
algcvga.5 𝑁 = (𝐶𝐴)
algfx.6 (𝑧𝑆 → ((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧))
Assertion
Ref Expression
algfx (𝐴𝑆 → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁)))
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.
StepHypRef Expression
1 algcvga.5 . . . 4 𝑁 = (𝐶𝐴)
2 algcvga.3 . . . . 5 𝐶:𝑆⟶ℕ0
32ffvelrni 5472 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
41, 3syl5eqel 2181 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
54nn0zd 8965 . 2 (𝐴𝑆𝑁 ∈ ℤ)
6 uzval 9120 . . . . . . 7 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑧 ∈ ℤ ∣ 𝑁𝑧})
76eleq2d 2164 . . . . . 6 (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) ↔ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
87pm5.32i 443 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
9 fveqeq2 5349 . . . . . . 7 (𝑚 = 𝑁 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑁) = (𝑅𝑁)))
109imbi2d 229 . . . . . 6 (𝑚 = 𝑁 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))))
11 fveqeq2 5349 . . . . . . 7 (𝑚 = 𝑘 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
1211imbi2d 229 . . . . . 6 (𝑚 = 𝑘 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑘) = (𝑅𝑁))))
13 fveqeq2 5349 . . . . . . 7 (𝑚 = (𝑘 + 1) → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
1413imbi2d 229 . . . . . 6 (𝑚 = (𝑘 + 1) → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
15 fveqeq2 5349 . . . . . . 7 (𝑚 = 𝐾 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝐾) = (𝑅𝑁)))
1615imbi2d 229 . . . . . 6 (𝑚 = 𝐾 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁))))
17 eqidd 2096 . . . . . . 7 (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))
1817a1i 9 . . . . . 6 (𝑁 ∈ ℤ → (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁)))
196eleq2d 2164 . . . . . . . . 9 (𝑁 ∈ ℤ → (𝑘 ∈ (ℤ𝑁) ↔ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
2019pm5.32i 443 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
21 eluznn0 9185 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
224, 21sylan 278 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
23 nn0uz 9152 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
24 algcvga.2 . . . . . . . . . . . . . . 15 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
25 0zd 8860 . . . . . . . . . . . . . . 15 (𝐴𝑆 → 0 ∈ ℤ)
26 id 19 . . . . . . . . . . . . . . 15 (𝐴𝑆𝐴𝑆)
27 algcvga.1 . . . . . . . . . . . . . . . 16 𝐹:𝑆𝑆
2827a1i 9 . . . . . . . . . . . . . . 15 (𝐴𝑆𝐹:𝑆𝑆)
2923, 24, 25, 26, 28algrp1 11455 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3022, 29syldan 277 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3123, 24, 25, 26, 28algrf 11454 . . . . . . . . . . . . . . . 16 (𝐴𝑆𝑅:ℕ0𝑆)
3231ffvelrnda 5473 . . . . . . . . . . . . . . 15 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
3322, 32syldan 277 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅𝑘) ∈ 𝑆)
34 algcvga.4 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
3527, 24, 2, 34, 1algcvga 11460 . . . . . . . . . . . . . . 15 (𝐴𝑆 → (𝑘 ∈ (ℤ𝑁) → (𝐶‘(𝑅𝑘)) = 0))
3635imp 123 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐶‘(𝑅𝑘)) = 0)
37 fveqeq2 5349 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐶𝑧) = 0 ↔ (𝐶‘(𝑅𝑘)) = 0))
38 fveq2 5340 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → (𝐹𝑧) = (𝐹‘(𝑅𝑘)))
39 id 19 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → 𝑧 = (𝑅𝑘))
4038, 39eqeq12d 2109 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐹𝑧) = 𝑧 ↔ (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
4137, 40imbi12d 233 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅𝑘) → (((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧) ↔ ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))))
42 algfx.6 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧))
4341, 42vtoclga 2699 . . . . . . . . . . . . . 14 ((𝑅𝑘) ∈ 𝑆 → ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
4433, 36, 43sylc 62 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))
4530, 44eqtrd 2127 . . . . . . . . . . . 12 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝑅𝑘))
4645eqeq1d 2103 . . . . . . . . . . 11 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅‘(𝑘 + 1)) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
4746biimprd 157 . . . . . . . . . 10 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
4847expcom 115 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
4948adantl 272 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5020, 49sylbir 134 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5150a2d 26 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → ((𝐴𝑆 → (𝑅𝑘) = (𝑅𝑁)) → (𝐴𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5210, 12, 14, 16, 18, 51uzind3 8958 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
538, 52sylbi 120 . . . 4 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
5453ex 114 . . 3 (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁))))
5554com3r 79 . 2 (𝐴𝑆 → (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁))))
565, 55mpd 13 1 (𝐴𝑆 → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  wne 2262  {crab 2374  {csn 3466   class class class wbr 3867   × cxp 4465  ccom 4471  wf 5045  cfv 5049  (class class class)co 5690  1st c1st 5947  0cc0 7447  1c1 7448   + caddc 7450   < clt 7619  cle 7620  0cn0 8771  cz 8848  cuz 9118  seqcseq 10000
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-seqfrec 10001
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator