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Theorem algfx 12055
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
32ffvelcdmi 5653 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
41, 3eqeltrid 2264 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
54nn0zd 9376 . 2 (𝐴𝑆𝑁 ∈ ℤ)
6 uzval 9533 . . . . . . 7 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑧 ∈ ℤ ∣ 𝑁𝑧})
76eleq2d 2247 . . . . . 6 (𝑁 ∈ ℤ → (𝐾 ∈ (ℤ𝑁) ↔ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
87pm5.32i 454 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
9 fveqeq2 5526 . . . . . . 7 (𝑚 = 𝑁 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑁) = (𝑅𝑁)))
109imbi2d 230 . . . . . 6 (𝑚 = 𝑁 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))))
11 fveqeq2 5526 . . . . . . 7 (𝑚 = 𝑘 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
1211imbi2d 230 . . . . . 6 (𝑚 = 𝑘 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝑘) = (𝑅𝑁))))
13 fveqeq2 5526 . . . . . . 7 (𝑚 = (𝑘 + 1) → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
1413imbi2d 230 . . . . . 6 (𝑚 = (𝑘 + 1) → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
15 fveqeq2 5526 . . . . . . 7 (𝑚 = 𝐾 → ((𝑅𝑚) = (𝑅𝑁) ↔ (𝑅𝐾) = (𝑅𝑁)))
1615imbi2d 230 . . . . . 6 (𝑚 = 𝐾 → ((𝐴𝑆 → (𝑅𝑚) = (𝑅𝑁)) ↔ (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁))))
17 eqidd 2178 . . . . . . 7 (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁))
1817a1i 9 . . . . . 6 (𝑁 ∈ ℤ → (𝐴𝑆 → (𝑅𝑁) = (𝑅𝑁)))
196eleq2d 2247 . . . . . . . . 9 (𝑁 ∈ ℤ → (𝑘 ∈ (ℤ𝑁) ↔ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
2019pm5.32i 454 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) ↔ (𝑁 ∈ ℤ ∧ 𝑘 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
21 eluznn0 9602 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
224, 21sylan 283 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → 𝑘 ∈ ℕ0)
23 nn0uz 9565 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
24 algcvga.2 . . . . . . . . . . . . . . 15 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
25 0zd 9268 . . . . . . . . . . . . . . 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 12049 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3022, 29syldan 282 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3123, 24, 25, 26, 28algrf 12048 . . . . . . . . . . . . . . . 16 (𝐴𝑆𝑅:ℕ0𝑆)
3231ffvelcdmda 5654 . . . . . . . . . . . . . . 15 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
3322, 32syldan 282 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅𝑘) ∈ 𝑆)
34 algcvga.4 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
3527, 24, 2, 34, 1algcvga 12054 . . . . . . . . . . . . . . 15 (𝐴𝑆 → (𝑘 ∈ (ℤ𝑁) → (𝐶‘(𝑅𝑘)) = 0))
3635imp 124 . . . . . . . . . . . . . 14 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐶‘(𝑅𝑘)) = 0)
37 fveqeq2 5526 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐶𝑧) = 0 ↔ (𝐶‘(𝑅𝑘)) = 0))
38 fveq2 5517 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → (𝐹𝑧) = (𝐹‘(𝑅𝑘)))
39 id 19 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑅𝑘) → 𝑧 = (𝑅𝑘))
4038, 39eqeq12d 2192 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑅𝑘) → ((𝐹𝑧) = 𝑧 ↔ (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
4137, 40imbi12d 234 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅𝑘) → (((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧) ↔ ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))))
42 algfx.6 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((𝐶𝑧) = 0 → (𝐹𝑧) = 𝑧))
4341, 42vtoclga 2805 . . . . . . . . . . . . . 14 ((𝑅𝑘) ∈ 𝑆 → ((𝐶‘(𝑅𝑘)) = 0 → (𝐹‘(𝑅𝑘)) = (𝑅𝑘)))
4433, 36, 43sylc 62 . . . . . . . . . . . . 13 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝐹‘(𝑅𝑘)) = (𝑅𝑘))
4530, 44eqtrd 2210 . . . . . . . . . . . 12 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → (𝑅‘(𝑘 + 1)) = (𝑅𝑘))
4645eqeq1d 2186 . . . . . . . . . . 11 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅‘(𝑘 + 1)) = (𝑅𝑁) ↔ (𝑅𝑘) = (𝑅𝑁)))
4746biimprd 158 . . . . . . . . . 10 ((𝐴𝑆𝑘 ∈ (ℤ𝑁)) → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁)))
4847expcom 116 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
4948adantl 277 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑘 ∈ (ℤ𝑁)) → (𝐴𝑆 → ((𝑅𝑘) = (𝑅𝑁) → (𝑅‘(𝑘 + 1)) = (𝑅𝑁))))
5020, 49sylbir 135 . . . . . . 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 9369 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
538, 52sylbi 121 . . . 4 ((𝑁 ∈ ℤ ∧ 𝐾 ∈ (ℤ𝑁)) → (𝐴𝑆 → (𝑅𝐾) = (𝑅𝑁)))
5453ex 115 . . 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 104   = wceq 1353  wcel 2148  wne 2347  {crab 2459  {csn 3594   class class class wbr 4005   × cxp 4626  ccom 4632  wf 5214  cfv 5218  (class class class)co 5878  1st c1st 6142  0cc0 7814  1c1 7815   + caddc 7817   < clt 7995  cle 7996  0cn0 9179  cz 9256  cuz 9531  seqcseq 10448
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-seqfrec 10449
This theorem is referenced by: (None)
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