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Theorem algfx 16521
Description: If 𝐹 reaches a fixed point when the countdown function 𝐢 reaches 0, 𝐹 remains fixed after 𝑁 steps. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
algcvga.1 𝐹:π‘†βŸΆπ‘†
algcvga.2 𝑅 = seq0((𝐹 ∘ 1st ), (β„•0 Γ— {𝐴}))
algcvga.3 𝐢:π‘†βŸΆβ„•0
algcvga.4 (𝑧 ∈ 𝑆 β†’ ((πΆβ€˜(πΉβ€˜π‘§)) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘§)) < (πΆβ€˜π‘§)))
algcvga.5 𝑁 = (πΆβ€˜π΄)
algfx.6 (𝑧 ∈ 𝑆 β†’ ((πΆβ€˜π‘§) = 0 β†’ (πΉβ€˜π‘§) = 𝑧))
Assertion
Ref Expression
algfx (𝐴 ∈ 𝑆 β†’ (𝐾 ∈ (β„€β‰₯β€˜π‘) β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘)))
Distinct variable groups:   𝑧,𝐢   𝑧,𝐹   𝑧,𝑅   𝑧,𝑆   𝑧,𝐾   𝑧,𝑁
Allowed substitution hint:   𝐴(𝑧)
Proof of Theorem algfx
Dummy variables π‘˜ π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algcvga.5 . . . 4 𝑁 = (πΆβ€˜π΄)
2 algcvga.3 . . . . 5 𝐢:π‘†βŸΆβ„•0
32ffvelcdmi 7084 . . . 4 (𝐴 ∈ 𝑆 β†’ (πΆβ€˜π΄) ∈ β„•0)
41, 3eqeltrid 2835 . . 3 (𝐴 ∈ 𝑆 β†’ 𝑁 ∈ β„•0)
54nn0zd 12588 . 2 (𝐴 ∈ 𝑆 β†’ 𝑁 ∈ β„€)
6 uzval 12828 . . . . . . 7 (𝑁 ∈ β„€ β†’ (β„€β‰₯β€˜π‘) = {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧})
76eleq2d 2817 . . . . . 6 (𝑁 ∈ β„€ β†’ (𝐾 ∈ (β„€β‰₯β€˜π‘) ↔ 𝐾 ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}))
87pm5.32i 573 . . . . 5 ((𝑁 ∈ β„€ ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘)) ↔ (𝑁 ∈ β„€ ∧ 𝐾 ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}))
9 fveqeq2 6899 . . . . . . 7 (π‘š = 𝑁 β†’ ((π‘…β€˜π‘š) = (π‘…β€˜π‘) ↔ (π‘…β€˜π‘) = (π‘…β€˜π‘)))
109imbi2d 339 . . . . . 6 (π‘š = 𝑁 β†’ ((𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘š) = (π‘…β€˜π‘)) ↔ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘) = (π‘…β€˜π‘))))
11 fveqeq2 6899 . . . . . . 7 (π‘š = π‘˜ β†’ ((π‘…β€˜π‘š) = (π‘…β€˜π‘) ↔ (π‘…β€˜π‘˜) = (π‘…β€˜π‘)))
1211imbi2d 339 . . . . . 6 (π‘š = π‘˜ β†’ ((𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘š) = (π‘…β€˜π‘)) ↔ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘˜) = (π‘…β€˜π‘))))
13 fveqeq2 6899 . . . . . . 7 (π‘š = (π‘˜ + 1) β†’ ((π‘…β€˜π‘š) = (π‘…β€˜π‘) ↔ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘)))
1413imbi2d 339 . . . . . 6 (π‘š = (π‘˜ + 1) β†’ ((𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘š) = (π‘…β€˜π‘)) ↔ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘))))
15 fveqeq2 6899 . . . . . . 7 (π‘š = 𝐾 β†’ ((π‘…β€˜π‘š) = (π‘…β€˜π‘) ↔ (π‘…β€˜πΎ) = (π‘…β€˜π‘)))
1615imbi2d 339 . . . . . 6 (π‘š = 𝐾 β†’ ((𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘š) = (π‘…β€˜π‘)) ↔ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘))))
17 eqidd 2731 . . . . . . 7 (𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘) = (π‘…β€˜π‘))
1817a1i 11 . . . . . 6 (𝑁 ∈ β„€ β†’ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘) = (π‘…β€˜π‘)))
196eleq2d 2817 . . . . . . . . 9 (𝑁 ∈ β„€ β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘) ↔ π‘˜ ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}))
2019pm5.32i 573 . . . . . . . 8 ((𝑁 ∈ β„€ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) ↔ (𝑁 ∈ β„€ ∧ π‘˜ ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}))
21 eluznn0 12905 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„•0 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ β„•0)
224, 21sylan 578 . . . . . . . . . . . . . 14 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ β„•0)
23 nn0uz 12868 . . . . . . . . . . . . . . 15 β„•0 = (β„€β‰₯β€˜0)
24 algcvga.2 . . . . . . . . . . . . . . 15 𝑅 = seq0((𝐹 ∘ 1st ), (β„•0 Γ— {𝐴}))
25 0zd 12574 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑆 β†’ 0 ∈ β„€)
26 id 22 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑆 β†’ 𝐴 ∈ 𝑆)
27 algcvga.1 . . . . . . . . . . . . . . . 16 𝐹:π‘†βŸΆπ‘†
2827a1i 11 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑆 β†’ 𝐹:π‘†βŸΆπ‘†)
2923, 24, 25, 26, 28algrp1 16515 . . . . . . . . . . . . . 14 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ β„•0) β†’ (π‘…β€˜(π‘˜ + 1)) = (πΉβ€˜(π‘…β€˜π‘˜)))
3022, 29syldan 589 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘…β€˜(π‘˜ + 1)) = (πΉβ€˜(π‘…β€˜π‘˜)))
3123, 24, 25, 26, 28algrf 16514 . . . . . . . . . . . . . . . 16 (𝐴 ∈ 𝑆 β†’ 𝑅:β„•0βŸΆπ‘†)
3231ffvelcdmda 7085 . . . . . . . . . . . . . . 15 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ β„•0) β†’ (π‘…β€˜π‘˜) ∈ 𝑆)
3322, 32syldan 589 . . . . . . . . . . . . . 14 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘…β€˜π‘˜) ∈ 𝑆)
34 algcvga.4 . . . . . . . . . . . . . . . 16 (𝑧 ∈ 𝑆 β†’ ((πΆβ€˜(πΉβ€˜π‘§)) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘§)) < (πΆβ€˜π‘§)))
3527, 24, 2, 34, 1algcvga 16520 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑆 β†’ (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (πΆβ€˜(π‘…β€˜π‘˜)) = 0))
3635imp 405 . . . . . . . . . . . . . 14 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΆβ€˜(π‘…β€˜π‘˜)) = 0)
37 fveqeq2 6899 . . . . . . . . . . . . . . . 16 (𝑧 = (π‘…β€˜π‘˜) β†’ ((πΆβ€˜π‘§) = 0 ↔ (πΆβ€˜(π‘…β€˜π‘˜)) = 0))
38 fveq2 6890 . . . . . . . . . . . . . . . . 17 (𝑧 = (π‘…β€˜π‘˜) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(π‘…β€˜π‘˜)))
39 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = (π‘…β€˜π‘˜) β†’ 𝑧 = (π‘…β€˜π‘˜))
4038, 39eqeq12d 2746 . . . . . . . . . . . . . . . 16 (𝑧 = (π‘…β€˜π‘˜) β†’ ((πΉβ€˜π‘§) = 𝑧 ↔ (πΉβ€˜(π‘…β€˜π‘˜)) = (π‘…β€˜π‘˜)))
4137, 40imbi12d 343 . . . . . . . . . . . . . . 15 (𝑧 = (π‘…β€˜π‘˜) β†’ (((πΆβ€˜π‘§) = 0 β†’ (πΉβ€˜π‘§) = 𝑧) ↔ ((πΆβ€˜(π‘…β€˜π‘˜)) = 0 β†’ (πΉβ€˜(π‘…β€˜π‘˜)) = (π‘…β€˜π‘˜))))
42 algfx.6 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑆 β†’ ((πΆβ€˜π‘§) = 0 β†’ (πΉβ€˜π‘§) = 𝑧))
4341, 42vtoclga 3565 . . . . . . . . . . . . . 14 ((π‘…β€˜π‘˜) ∈ 𝑆 β†’ ((πΆβ€˜(π‘…β€˜π‘˜)) = 0 β†’ (πΉβ€˜(π‘…β€˜π‘˜)) = (π‘…β€˜π‘˜)))
4433, 36, 43sylc 65 . . . . . . . . . . . . 13 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜(π‘…β€˜π‘˜)) = (π‘…β€˜π‘˜))
4530, 44eqtrd 2770 . . . . . . . . . . . 12 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘˜))
4645eqeq1d 2732 . . . . . . . . . . 11 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ ((π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘) ↔ (π‘…β€˜π‘˜) = (π‘…β€˜π‘)))
4746biimprd 247 . . . . . . . . . 10 ((𝐴 ∈ 𝑆 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ ((π‘…β€˜π‘˜) = (π‘…β€˜π‘) β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘)))
4847expcom 412 . . . . . . . . 9 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ (𝐴 ∈ 𝑆 β†’ ((π‘…β€˜π‘˜) = (π‘…β€˜π‘) β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘))))
4948adantl 480 . . . . . . . 8 ((𝑁 ∈ β„€ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (𝐴 ∈ 𝑆 β†’ ((π‘…β€˜π‘˜) = (π‘…β€˜π‘) β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘))))
5020, 49sylbir 234 . . . . . . 7 ((𝑁 ∈ β„€ ∧ π‘˜ ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}) β†’ (𝐴 ∈ 𝑆 β†’ ((π‘…β€˜π‘˜) = (π‘…β€˜π‘) β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘))))
5150a2d 29 . . . . . 6 ((𝑁 ∈ β„€ ∧ π‘˜ ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}) β†’ ((𝐴 ∈ 𝑆 β†’ (π‘…β€˜π‘˜) = (π‘…β€˜π‘)) β†’ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜(π‘˜ + 1)) = (π‘…β€˜π‘))))
5210, 12, 14, 16, 18, 51uzind3 12660 . . . . 5 ((𝑁 ∈ β„€ ∧ 𝐾 ∈ {𝑧 ∈ β„€ ∣ 𝑁 ≀ 𝑧}) β†’ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘)))
538, 52sylbi 216 . . . 4 ((𝑁 ∈ β„€ ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘)) β†’ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘)))
5453ex 411 . . 3 (𝑁 ∈ β„€ β†’ (𝐾 ∈ (β„€β‰₯β€˜π‘) β†’ (𝐴 ∈ 𝑆 β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘))))
5554com3r 87 . 2 (𝐴 ∈ 𝑆 β†’ (𝑁 ∈ β„€ β†’ (𝐾 ∈ (β„€β‰₯β€˜π‘) β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘))))
565, 55mpd 15 1 (𝐴 ∈ 𝑆 β†’ (𝐾 ∈ (β„€β‰₯β€˜π‘) β†’ (π‘…β€˜πΎ) = (π‘…β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  {crab 3430  {csn 4627   class class class wbr 5147   Γ— cxp 5673   ∘ ccom 5679  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  seqcseq 13970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-seq 13971
This theorem is referenced by: (None)
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