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Theorem ackvalsuc1mpt 47451
Description: The Ackermann function at a successor of the first argument as a mapping of the second argument. (Contributed by Thierry Arnoux, 28-Apr-2024.) (Revised by AV, 4-May-2024.)
Assertion
Ref Expression
ackvalsuc1mpt (𝑀 ∈ β„•0 β†’ (Ackβ€˜(𝑀 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
Distinct variable group:   𝑛,𝑀
Proof of Theorem ackvalsuc1mpt
Dummy variables 𝑓 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ack 47433 . . 3 Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))
21fveq1i 6891 . 2 (Ackβ€˜(𝑀 + 1)) = (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))β€˜(𝑀 + 1))
3 nn0uz 12868 . . . 4 β„•0 = (β„€β‰₯β€˜0)
4 id 22 . . . 4 (𝑀 ∈ β„•0 β†’ 𝑀 ∈ β„•0)
5 eqid 2730 . . . 4 (𝑀 + 1) = (𝑀 + 1)
61eqcomi 2739 . . . . . 6 seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖))) = Ack
76fveq1i 6891 . . . . 5 (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))β€˜π‘€) = (Ackβ€˜π‘€)
87a1i 11 . . . 4 (𝑀 ∈ β„•0 β†’ (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))β€˜π‘€) = (Ackβ€˜π‘€))
9 eqidd 2731 . . . . 5 (𝑀 ∈ β„•0 β†’ (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)) = (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))
10 nn0p1gt0 12505 . . . . . . . . . . 11 (𝑀 ∈ β„•0 β†’ 0 < (𝑀 + 1))
1110gt0ne0d 11782 . . . . . . . . . 10 (𝑀 ∈ β„•0 β†’ (𝑀 + 1) β‰  0)
1211adantr 479 . . . . . . . . 9 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ (𝑀 + 1) β‰  0)
13 neeq1 3001 . . . . . . . . . 10 (𝑖 = (𝑀 + 1) β†’ (𝑖 β‰  0 ↔ (𝑀 + 1) β‰  0))
1413adantl 480 . . . . . . . . 9 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ (𝑖 β‰  0 ↔ (𝑀 + 1) β‰  0))
1512, 14mpbird 256 . . . . . . . 8 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ 𝑖 β‰  0)
1615neneqd 2943 . . . . . . 7 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ Β¬ 𝑖 = 0)
1716iffalsed 4538 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖) = 𝑖)
18 simpr 483 . . . . . 6 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ 𝑖 = (𝑀 + 1))
1917, 18eqtrd 2770 . . . . 5 ((𝑀 ∈ β„•0 ∧ 𝑖 = (𝑀 + 1)) β†’ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖) = (𝑀 + 1))
20 peano2nn0 12516 . . . . 5 (𝑀 ∈ β„•0 β†’ (𝑀 + 1) ∈ β„•0)
219, 19, 20, 20fvmptd 7004 . . . 4 (𝑀 ∈ β„•0 β†’ ((𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖))β€˜(𝑀 + 1)) = (𝑀 + 1))
223, 4, 5, 8, 21seqp1d 13987 . . 3 (𝑀 ∈ β„•0 β†’ (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))β€˜(𝑀 + 1)) = ((Ackβ€˜π‘€)(𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1)))(𝑀 + 1)))
23 eqidd 2731 . . . 4 (𝑀 ∈ β„•0 β†’ (𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))) = (𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))))
24 fveq2 6890 . . . . . . . 8 (𝑓 = (Ackβ€˜π‘€) β†’ (IterCompβ€˜π‘“) = (IterCompβ€˜(Ackβ€˜π‘€)))
2524fveq1d 6892 . . . . . . 7 (𝑓 = (Ackβ€˜π‘€) β†’ ((IterCompβ€˜π‘“)β€˜(𝑛 + 1)) = ((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1)))
2625fveq1d 6892 . . . . . 6 (𝑓 = (Ackβ€˜π‘€) β†’ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1) = (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1))
2726mpteq2dv 5249 . . . . 5 (𝑓 = (Ackβ€˜π‘€) β†’ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
2827ad2antrl 724 . . . 4 ((𝑀 ∈ β„•0 ∧ (𝑓 = (Ackβ€˜π‘€) ∧ 𝑗 = (𝑀 + 1))) β†’ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
29 fvexd 6905 . . . 4 (𝑀 ∈ β„•0 β†’ (Ackβ€˜π‘€) ∈ V)
30 ovexd 7446 . . . 4 (𝑀 ∈ β„•0 β†’ (𝑀 + 1) ∈ V)
31 nn0ex 12482 . . . . . 6 β„•0 ∈ V
3231mptex 7226 . . . . 5 (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)) ∈ V
3332a1i 11 . . . 4 (𝑀 ∈ β„•0 β†’ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)) ∈ V)
3423, 28, 29, 30, 33ovmpod 7562 . . 3 (𝑀 ∈ β„•0 β†’ ((Ackβ€˜π‘€)(𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1)))(𝑀 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
3522, 34eqtrd 2770 . 2 (𝑀 ∈ β„•0 β†’ (seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜π‘“)β€˜(𝑛 + 1))β€˜1))), (𝑖 ∈ β„•0 ↦ if(𝑖 = 0, (𝑛 ∈ β„•0 ↦ (𝑛 + 1)), 𝑖)))β€˜(𝑀 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
362, 35eqtrid 2782 1 (𝑀 ∈ β„•0 β†’ (Ackβ€˜(𝑀 + 1)) = (𝑛 ∈ β„•0 ↦ (((IterCompβ€˜(Ackβ€˜π‘€))β€˜(𝑛 + 1))β€˜1)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472  ifcif 4527   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  0cc0 11112  1c1 11113   + caddc 11115  β„•0cn0 12476  seqcseq 13970  IterCompcitco 47430  Ackcack 47431
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 5284  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-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-seq 13971  df-ack 47433
This theorem is referenced by:  ackvalsuc1  47452  ackval1  47454  ackval2  47455  ackval3  47456  ackendofnn0  47457
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