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Theorem ackvalsucsucval 47374
Description: The Ackermann function at the successors. This is the third equation of PΓ©ter's definition of the Ackermann function. (Contributed by AV, 8-May-2024.)
Assertion
Ref Expression
ackvalsucsucval ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜(𝑁 + 1)) = ((Ackβ€˜π‘€)β€˜((Ackβ€˜(𝑀 + 1))β€˜π‘)))
Proof of Theorem ackvalsucsucval
StepHypRef Expression
1 peano2nn0 12512 . . 3 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
2 ackvalsuc1 47365 . . 3 ((𝑀 ∈ β„•0 ∧ (𝑁 + 1) ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜(𝑁 + 1)) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1))β€˜1))
31, 2sylan2 594 . 2 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜(𝑁 + 1)) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1))β€˜1))
4 fvexd 6907 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€) ∈ V)
51adantl 483 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 + 1) ∈ β„•0)
6 eqidd 2734 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))
7 itcovalsucov 47354 . . . . 5 (((Ackβ€˜π‘€) ∈ V ∧ (𝑁 + 1) ∈ β„•0 ∧ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))))
84, 5, 6, 7syl3anc 1372 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))))
98fveq1d 6894 . . 3 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1))β€˜1) = (((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))β€˜1))
10 ackfnnn0 47371 . . . . . . 7 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€) Fn β„•0)
1110adantr 482 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€) Fn β„•0)
12 nn0ex 12478 . . . . . . . . 9 β„•0 ∈ V
1312a1i 11 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ β„•0 ∈ V)
14 ackendofnn0 47370 . . . . . . . . 9 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
1514adantr 482 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
16 simpr 486 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ 𝑁 ∈ β„•0)
1713, 15, 16itcovalendof 47355 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘):β„•0βŸΆβ„•0)
1817ffnd 6719 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) Fn β„•0)
1917frnd 6726 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ran ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) βŠ† β„•0)
20 fnco 6668 . . . . . 6 (((Ackβ€˜π‘€) Fn β„•0 ∧ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) Fn β„•0 ∧ ran ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) βŠ† β„•0) β†’ ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) Fn β„•0)
2111, 18, 19, 20syl3anc 1372 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) Fn β„•0)
22 eqidd 2734 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘))
23 itcovalsucov 47354 . . . . . . 7 (((Ackβ€˜π‘€) ∈ V ∧ 𝑁 ∈ β„•0 ∧ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)))
244, 16, 22, 23syl3anc 1372 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)))
2524fneq1d 6643 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) Fn β„•0 ↔ ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) Fn β„•0))
2621, 25mpbird 257 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) Fn β„•0)
27 1nn0 12488 . . . 4 1 ∈ β„•0
28 fvco2 6989 . . . 4 ((((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) Fn β„•0 ∧ 1 ∈ β„•0) β†’ (((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))β€˜1) = ((Ackβ€˜π‘€)β€˜(((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1)))
2926, 27, 28sylancl 587 . . 3 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))β€˜1) = ((Ackβ€˜π‘€)β€˜(((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1)))
309, 29eqtrd 2773 . 2 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1))β€˜1) = ((Ackβ€˜π‘€)β€˜(((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1)))
31 ackvalsuc1 47365 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜π‘) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1))
3231eqcomd 2739 . . 3 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1) = ((Ackβ€˜(𝑀 + 1))β€˜π‘))
3332fveq2d 6896 . 2 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜π‘€)β€˜(((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1)) = ((Ackβ€˜π‘€)β€˜((Ackβ€˜(𝑀 + 1))β€˜π‘)))
343, 30, 333eqtrd 2777 1 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜(𝑁 + 1)) = ((Ackβ€˜π‘€)β€˜((Ackβ€˜(𝑀 + 1))β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βŠ† wss 3949  ran crn 5678   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1c1 11111   + caddc 11113  β„•0cn0 12472  IterCompcitco 47343  Ackcack 47344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-itco 47345  df-ack 47346
This theorem is referenced by:  ackval41a  47380  ackval42  47382
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