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Theorem ialgfx 10812
 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}), 𝑆)
algcvga.3 𝐶:𝑆⟶ℕ0
algcvga.4 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
algcvga.5 𝑁 = (𝐶𝐴)
ialgcvga.s 𝑆𝑉
algfx.6 (𝑧𝑆 → ((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧))
Assertion
Ref Expression
ialgfx (𝐴𝑆 → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁)))
Distinct variable groups:   𝑧,𝐶   𝑧,𝐹   𝑧,𝑅   𝑧,𝑆   𝑧,𝐾   𝑧,𝑁
Allowed substitution hints:   𝐴(𝑧)   𝑉(𝑧)

Proof of Theorem ialgfx
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 5376 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
41, 3syl5eqel 2169 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
54nn0zd 8760 . 2 (𝐴𝑆𝑁 ∈ ℤ)
6 uzval 8914 . . . . . . 7 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑧 ∈ ℤ ∣ 𝑁𝑧})
76eleq2d 2152 . . . . . 6 (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) ↔ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
87pm5.32i 442 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
9 fveq2 5251 . . . . . . . 8 (𝑚 = 𝑁 → (𝑅𝑚) = (𝑅𝑁))
109eqeq1d 2091 . . . . . . 7 (𝑚 = 𝑁 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑁) = (𝑅𝑁)))
1110imbi2d 228 . . . . . 6 (𝑚 = 𝑁 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))))
12 fveq2 5251 . . . . . . . 8 (𝑚 = 𝑘 → (𝑅𝑚) = (𝑅𝑘))
1312eqeq1d 2091 . . . . . . 7 (𝑚 = 𝑘 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
1413imbi2d 228 . . . . . 6 (𝑚 = 𝑘 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑘) = (𝑅𝑁))))
15 fveq2 5251 . . . . . . . 8 (𝑚 = (𝑘 + 1) → (𝑅𝑚) = (𝑅‘(𝑘 + 1)))
1615eqeq1d 2091 . . . . . . 7 (𝑚 = (𝑘 + 1) → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
1716imbi2d 228 . . . . . 6 (𝑚 = (𝑘 + 1) → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
18 fveq2 5251 . . . . . . . 8 (𝑚 = 𝐾 → (𝑅𝑚) = (𝑅𝐾))
1918eqeq1d 2091 . . . . . . 7 (𝑚 = 𝐾 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝐾) = (𝑅𝑁)))
2019imbi2d 228 . . . . . 6 (𝑚 = 𝐾 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁))))
21 eqidd 2084 . . . . . . 7 (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))
2221a1i 9 . . . . . 6 (𝑁 ∈ ℤ → (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁)))
236eleq2d 2152 . . . . . . . . 9 (𝑁 ∈ ℤ → (𝑘 ∈ (ℤ𝑁) ↔ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
2423pm5.32i 442 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
25 eluznn0 8979 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
264, 25sylan 277 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
27 nn0uz 8946 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
28 algcvga.2 . . . . . . . . . . . . . . 15 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}), 𝑆)
29 0zd 8656 . . . . . . . . . . . . . . 15 (𝐴𝑆 → 0 ∈ ℤ)
30 id 19 . . . . . . . . . . . . . . 15 (𝐴𝑆𝐴𝑆)
31 algcvga.1 . . . . . . . . . . . . . . . 16 𝐹:𝑆𝑆
3231a1i 9 . . . . . . . . . . . . . . 15 (𝐴𝑆𝐹:𝑆𝑆)
33 ialgcvga.s . . . . . . . . . . . . . . . 16 𝑆𝑉
3433a1i 9 . . . . . . . . . . . . . . 15 (𝐴𝑆𝑆𝑉)
3527, 28, 29, 30, 32, 34ialgrp1 10806 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3626, 35syldan 276 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3727, 28, 29, 30, 32, 34ialgrf 10805 . . . . . . . . . . . . . . . 16 (𝐴𝑆𝑅:ℕ0𝑆)
3837ffvelrnda 5377 . . . . . . . . . . . . . . 15 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
3926, 38syldan 276 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅𝑘) ∈ 𝑆)
40 algcvga.4 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
4131, 28, 2, 40, 1, 33ialgcvga 10811 . . . . . . . . . . . . . . 15 (𝐴𝑆 → (𝑘 ∈ (ℤ𝑁) → (𝐶‘(𝑅𝑘)) = 0))
4241imp 122 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐶‘(𝑅𝑘)) = 0)
43 fveq2 5251 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → (𝐶𝑧) = (𝐶‘(𝑅𝑘)))
4443eqeq1d 2091 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐶𝑧) = 0 ↔ (𝐶‘(𝑅𝑘)) = 0))
45 fveq2 5251 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → (𝐹𝑧) = (𝐹‘(𝑅𝑘)))
46 id 19 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → 𝑧 = (𝑅𝑘))
4745, 46eqeq12d 2097 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐹𝑧) = 𝑧 ↔ (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
4844, 47imbi12d 232 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅𝑘) → (((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧) ↔ ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))))
49 algfx.6 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧))
5048, 49vtoclga 2675 . . . . . . . . . . . . . 14 ((𝑅𝑘) ∈ 𝑆 → ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
5139, 42, 50sylc 61 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))
5236, 51eqtrd 2115 . . . . . . . . . . . 12 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝑅𝑘))
5352eqeq1d 2091 . . . . . . . . . . 11 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅‘(𝑘 + 1)) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
5453biimprd 156 . . . . . . . . . 10 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
5554expcom 114 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5655adantl 271 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5724, 56sylbir 133 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5857a2d 26 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → ((𝐴𝑆 → (𝑅𝑘) = (𝑅𝑁)) → (𝐴𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5911, 14, 17, 20, 22, 58uzind3 8753 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
608, 59sylbi 119 . . . 4 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
6160ex 113 . . 3 (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁))))
6261com3r 78 . 2 (𝐴𝑆 → (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁))))
635, 62mpd 13 1 (𝐴𝑆 → (𝐾 ∈ (ℤ𝑁) → (𝑅𝐾) = (𝑅𝑁)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434   ≠ wne 2249  {crab 2357  {csn 3422   class class class wbr 3811   × cxp 4397   ∘ ccom 4403  ⟶wf 4963  ‘cfv 4967  (class class class)co 5589  1st c1st 5842  0cc0 7251  1c1 7252   + caddc 7254   < clt 7423   ≤ cle 7424  ℕ0cn0 8563  ℤcz 8644  ℤ≥cuz 8912  seqcseq 9738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-apti 7361  ax-pre-ltadd 7362 This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913  df-iseq 9739 This theorem is referenced by: (None)
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