Theorem ackendofnn0 48677

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 6861	. . 3 ⊢ (𝑥 = 0 → (Ack‘𝑥) = (Ack‘0))
2	1	feq1d 6673	. 2 ⊢ (𝑥 = 0 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘0):ℕ0⟶ℕ0))
3		fveq2 6861	. . 3 ⊢ (𝑥 = 𝑦 → (Ack‘𝑥) = (Ack‘𝑦))
4	3	feq1d 6673	. 2 ⊢ (𝑥 = 𝑦 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘𝑦):ℕ0⟶ℕ0))
5		fveq2 6861	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (Ack‘𝑥) = (Ack‘(𝑦 + 1)))
6	5	feq1d 6673	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0))
7		fveq2 6861	. . 3 ⊢ (𝑥 = 𝑀 → (Ack‘𝑥) = (Ack‘𝑀))
8	7	feq1d 6673	. 2 ⊢ (𝑥 = 𝑀 → ((Ack‘𝑥):ℕ0⟶ℕ0 ↔ (Ack‘𝑀):ℕ0⟶ℕ0))
9		ackval0 48673	. . 3 ⊢ (Ack‘0) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 1))
10		peano2nn0 12489	. . 3 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
11	9, 10	fmpti 7087	. 2 ⊢ (Ack‘0):ℕ0⟶ℕ0
12		nn0ex 12455	. . . . . . . 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 481	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + 1) ∈ ℕ0)
16	13, 14, 15	itcovalendof 48662	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → ((IterComp‘(Ack‘𝑦))‘(𝑛 + 1)):ℕ0⟶ℕ0)
17		1nn0 12465	. . . . . 6 ⊢ 1 ∈ ℕ0
18		ffvelcdm 7056	. . . . . 6 ⊢ ((((IterComp‘(Ack‘𝑦))‘(𝑛 + 1)):ℕ0⟶ℕ0 ∧ 1 ∈ ℕ0) → (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1) ∈ ℕ0)
19	16, 17, 18	sylancl 586	. . . . 5 ⊢ (((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) ∧ 𝑛 ∈ ℕ0) → (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1) ∈ ℕ0)
20		eqid 2730	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1))
21	19, 20	fmptd 7089	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)):ℕ0⟶ℕ0)
22		ackvalsuc1mpt 48671	. . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (Ack‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)))
23	22	adantr 480	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (Ack‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘𝑦))‘(𝑛 + 1))‘1)))
24	23	feq1d 6673	. . . 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 257	. . 3 ⊢ ((𝑦 ∈ ℕ0 ∧ (Ack‘𝑦):ℕ0⟶ℕ0) → (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0)
26	25	ex 412	. 2 ⊢ (𝑦 ∈ ℕ0 → ((Ack‘𝑦):ℕ0⟶ℕ0 → (Ack‘(𝑦 + 1)):ℕ0⟶ℕ0))
27	2, 4, 6, 8, 11, 26	nn0ind 12636	1 ⊢ (𝑀 ∈ ℕ0 → (Ack‘𝑀):ℕ0⟶ℕ0)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  ℕ₀cn0 12449  IterCompcitco 48650  Ackcack 48651

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974  df-itco 48652  df-ack 48653

This theorem is referenced by:  ackfnnn0  48678  ackvalsucsucval  48681