Theorem ackendofnn0 45463

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 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)

Proof of Theorem ackendofnn0

Dummy variables 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6658	. . 3 ⊢ (𝑥 = 0 → (Ack‘𝑥) = (Ack‘0))
2	1	feq1d 6483	. 2 ⊢ (𝑥 = 0 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘0):ℕ0⟶ℕ0))
3		fveq2 6658	. . 3 ⊢ (𝑥 = 𝑦 → (Ack‘𝑥) = (Ack‘𝑦))
4	3	feq1d 6483	. 2 ⊢ (𝑥 = 𝑦 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘𝑦):ℕ0⟶ℕ0))
5		fveq2 6658	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (Ack‘𝑥) = (Ack‘(𝑦 + 1)))
6	5	feq1d 6483	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0))
7		fveq2 6658	. . 3 ⊢ (𝑥 = 𝑀 → (Ack‘𝑥) = (Ack‘𝑀))
8	7	feq1d 6483	. 2 ⊢ (𝑥 = 𝑀 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘𝑀):ℕ0⟶ℕ0))
9		ackval0 45459	. . 3 ⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
10		peano2nn0 11974	. . 3 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
11	9, 10	fmpti 6867	. 2 ⊢ (Ack‘0):ℕ0⟶ℕ0
12		nn0ex 11940	. . . . . . . 8 ⊢ ℕ0 ∈ V
13	12	a1i 11	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → ℕ0 ∈ V)
14		simplr 768	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → (Ack‘𝑦):ℕ0⟶ℕ0)
15	10	adantl 485	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
16	13, 14, 15	itcovalendof 45448	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → ((IterComp‘(Ack‘𝑦))‘(𝑛 + 1)):ℕ0⟶ℕ0)
17		1nn0 11950	. . . . . 6 ⊢ 1 ∈ ℕ0
18		ffvelrn 6840	. . . . . 6 ⊢ ((((IterComp‘(Ack‘𝑦))‘(𝑛 + 1)):ℕ0⟶ℕ0 ∧ 1 ∈ ℕ0) → (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1) ∈ ℕ0)
19	16, 17, 18	sylancl 589	. . . . 5 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1) ∈ ℕ0)
20		eqid 2758	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1))
21	19, 20	fmptd 6869	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)):ℕ0⟶ℕ0)
22		ackvalsuc1mpt 45457	. . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (Ack‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)))
23	22	adantr 484	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (Ack‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)))
24	23	feq1d 6483	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → ((Ack‘(𝑦 + 1)):ℕ0⟶ℕ0 ↔ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)):ℕ0⟶ℕ0))
25	21, 24	mpbird 260	. . 3 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0)
26	25	ex 416	. 2 ⊢ (𝑦 ∈ ℕ0 → ((Ack‘𝑦):ℕ0⟶ℕ0 → (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0))
27	2, 4, 6, 8, 11, 26	nn0ind 12116	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576   + caddc 10578  ℕ₀cn0 11934  IterCompcitco 45436  Ackcack 45437

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-seq 13419  df-itco 45438  df-ack 45439

This theorem is referenced by:  ackfnnn0  45464  ackvalsucsucval  45467