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Theorem ackvalsucsucval 47364
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 12511 . . 3 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
2 ackvalsuc1 47355 . . 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 593 . 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 6906 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€) ∈ V)
51adantl 482 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 + 1) ∈ β„•0)
6 eqidd 2733 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)))
7 itcovalsucov 47344 . . . . 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 1371 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜((𝑁 + 1) + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))))
98fveq1d 6893 . . 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 47361 . . . . . . 7 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€) Fn β„•0)
1110adantr 481 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€) Fn β„•0)
12 nn0ex 12477 . . . . . . . . 9 β„•0 ∈ V
1312a1i 11 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ β„•0 ∈ V)
14 ackendofnn0 47360 . . . . . . . . 9 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
1514adantr 481 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (Ackβ€˜π‘€):β„•0βŸΆβ„•0)
16 simpr 485 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ 𝑁 ∈ β„•0)
1713, 15, 16itcovalendof 47345 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘):β„•0βŸΆβ„•0)
1817ffnd 6718 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) Fn β„•0)
1917frnd 6725 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ran ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) βŠ† β„•0)
20 fnco 6667 . . . . . 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 1371 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)) Fn β„•0)
22 eqidd 2733 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘))
23 itcovalsucov 47344 . . . . . . 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 1371 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1)) = ((Ackβ€˜π‘€) ∘ ((IterCompβ€˜(Ackβ€˜π‘€))β€˜π‘)))
2524fneq1d 6642 . . . . 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 12487 . . . 4 1 ∈ β„•0
28 fvco2 6988 . . . 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 586 . . 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 2772 . 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 47355 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((Ackβ€˜(𝑀 + 1))β€˜π‘) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1))
3231eqcomd 2738 . . 3 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑁 + 1))β€˜1) = ((Ackβ€˜(𝑀 + 1))β€˜π‘))
3332fveq2d 6895 . 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 2776 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  ran crn 5677   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  1c1 11110   + caddc 11112  β„•0cn0 12471  IterCompcitco 47333  Ackcack 47334
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-seq 13966  df-itco 47335  df-ack 47336
This theorem is referenced by:  ackval41a  47370  ackval42  47372
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