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Theorem ackvalsucsucval 47463
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 12518 . . 3 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
2 ackvalsuc1 47454 . . 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 591 . 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 480 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 + 1) ∈ β„•0)
6 eqidd 2731 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))
7 itcovalsucov 47443 . . . . 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 1369 . . . 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 47460 . . . . . . 7 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€) Fn β„•0)
1110adantr 479 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€) Fn β„•0)
12 nn0ex 12484 . . . . . . . . 9 β„•0 ∈ V
1312a1i 11 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ β„•0 ∈ V)
14 ackendofnn0 47459 . . . . . . . . 9 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
1514adantr 479 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
16 simpr 483 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ 𝑁 ∈ β„•0)
1713, 15, 16itcovalendof 47444 . . . . . . 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 1369 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) Fn β„•0)
22 eqidd 2731 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘))
23 itcovalsucov 47443 . . . . . . 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 1369 . . . . . 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 256 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) Fn β„•0)
27 1nn0 12494 . . . 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 584 . . 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 2770 . 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 47454 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜π‘) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1))
3231eqcomd 2736 . . 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 2774 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3949  ran crn 5678   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  1c1 11115   + caddc 11117  β„•0cn0 12478  IterCompcitco 47432  Ackcack 47433
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-inf2 9640  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
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 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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-n0 12479  df-z 12565  df-uz 12829  df-seq 13973  df-itco 47434  df-ack 47435
This theorem is referenced by:  ackval41a  47469  ackval42  47471
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