Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ackendofnn0 Structured version   Visualization version   GIF version
Theorem ackendofnn0 46923
Description: The Ackermann function at any nonnegative integer is an endofunction on the nonnegative integers. (Contributed by AV, 8-May-2024.)
Assertion
Ref Expression
ackendofnn0 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
Proof of Theorem ackendofnn0
Dummy variables π‘₯ 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6862 . . 3 (π‘₯ = 0 β†’ (Ackβ€˜π‘₯) = (Ackβ€˜0))
21feq1d 6673 . 2 (π‘₯ = 0 β†’ ((Ackβ€˜π‘₯):β„•0βŸΆβ„•0 ↔ (Ackβ€˜0):β„•0βŸΆβ„•0))
3 fveq2 6862 . . 3 (π‘₯ = 𝑦 β†’ (Ackβ€˜π‘₯) = (Ackβ€˜π‘¦))
43feq1d 6673 . 2 (π‘₯ = 𝑦 β†’ ((Ackβ€˜π‘₯):β„•0βŸΆβ„•0 ↔ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0))
5 fveq2 6862 . . 3 (π‘₯ = (𝑦 + 1) β†’ (Ackβ€˜π‘₯) = (Ackβ€˜(𝑦 + 1)))
65feq1d 6673 . 2 (π‘₯ = (𝑦 + 1) β†’ ((Ackβ€˜π‘₯):β„•0βŸΆβ„•0 ↔ (Ackβ€˜(𝑦 + 1)):β„•0βŸΆβ„•0))
7 fveq2 6862 . . 3 (π‘₯ = 𝑀 β†’ (Ackβ€˜π‘₯) = (Ackβ€˜π‘€))
87feq1d 6673 . 2 (π‘₯ = 𝑀 β†’ ((Ackβ€˜π‘₯):β„•0βŸΆβ„•0 ↔ (Ackβ€˜π‘€):β„•0βŸΆβ„•0))
9 ackval0 46919 . . 3 (Ackβ€˜0) = (𝑛 ∈ β„•0 ↦ (𝑛 + 1))
10 peano2nn0 12477 . . 3 (𝑛 ∈ β„•0 β†’ (𝑛 + 1) ∈ β„•0)
119, 10fmpti 7080 . 2 (Ackβ€˜0):β„•0βŸΆβ„•0
12 nn0ex 12443 . . . . . . . 8 β„•0 ∈ V
1312a1i 11 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) ∧ 𝑛 ∈ β„•0) β†’ β„•0 ∈ V)
14 simplr 767 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) ∧ 𝑛 ∈ β„•0) β†’ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0)
1510adantl 482 . . . . . . 7 (((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 + 1) ∈ β„•0)
1613, 14, 15itcovalendof 46908 . . . . . 6 (((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) ∧ 𝑛 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1)):β„•0βŸΆβ„•0)
17 1nn0 12453 . . . . . 6 1 ∈ β„•0
18 ffvelcdm 7052 . . . . . 6 ((((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1)):β„•0βŸΆβ„•0 ∧ 1 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1) ∈ β„•0)
1916, 17, 18sylancl 586 . . . . 5 (((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) ∧ 𝑛 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1) ∈ β„•0)
20 eqid 2731 . . . . 5 (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1))
2119, 20fmptd 7082 . . . 4 ((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) β†’ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1)):β„•0βŸΆβ„•0)
22 ackvalsuc1mpt 46917 . . . . . 6 (𝑦 ∈ β„•0 β†’ (Ackβ€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1)))
2322adantr 481 . . . . 5 ((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) β†’ (Ackβ€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1)))
2423feq1d 6673 . . . 4 ((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) β†’ ((Ackβ€˜(𝑦 + 1)):β„•0βŸΆβ„•0 ↔ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘¦))β€˜(𝑛 + 1))β€˜1)):β„•0βŸΆβ„•0))
2521, 24mpbird 256 . . 3 ((𝑦 ∈ β„•0 ∧ (Ackβ€˜π‘¦):β„•0βŸΆβ„•0) β†’ (Ackβ€˜(𝑦 + 1)):β„•0βŸΆβ„•0)
2625ex 413 . 2 (𝑦 ∈ β„•0 β†’ ((Ackβ€˜π‘¦):β„•0βŸΆβ„•0 β†’ (Ackβ€˜(𝑦 + 1)):β„•0βŸΆβ„•0))
272, 4, 6, 8, 11, 26nn0ind 12622 1 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3459   ↦ cmpt 5208  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  0cc0 11075  1c1 11076   + caddc 11078  β„•0cn0 12437  IterCompcitco 46896  Ackcack 46897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-seq 13932  df-itco 46898  df-ack 46899
This theorem is referenced by:  ackfnnn0  46924  ackvalsucsucval  46927
  Copyright terms: Public domain W3C validator