Theorem ackval1 48843

Description: The Ackermann function at 1. (Contributed by AV, 4-May-2024.)

Assertion

Ref	Expression
ackval1	⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Proof of Theorem ackval1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1e0p1 12640	. . 3 ⊢ 1 = (0 + 1)
2	1	fveq2i 6834	. 2 ⊢ (Ack‘1) = (Ack‘(0 + 1))
3		0nn0 12407	. . 3 ⊢ 0 ∈ ℕ0
4		ackvalsuc1mpt 48840	. . 3 ⊢ (0 ∈ ℕ0 → (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)))
5	3, 4	ax-mp 5	. 2 ⊢ (Ack‘(0 + 1)) = (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1))
6		peano2nn0 12432	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7		1nn0 12408	. . . . . . 7 ⊢ 1 ∈ ℕ0
8		ackval0 48842	. . . . . . . 8 ⊢ (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1))
9	8	itcovalpc 48834	. . . . . . 7 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 1 ∈ ℕ0) → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
10	6, 7, 9	sylancl 586	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))))
11		nn0cn 12402	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
13	12	mullidd 11141	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1))
14	13	oveq2d 7371	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1)))
15	14	mpteq2dv 5189	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
16	10, 15	eqtrd 2768	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
17	16	fveq1d 6833	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1))
18		eqidd 2734	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
19		oveq1 7362	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
20	19	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑖 = 1) → (𝑖 + (𝑛 + 1)) = (1 + (𝑛 + 1)))
21	7	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
22		ovexd 7390	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V)
23	18, 20, 21, 22	fvmptd 6945	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1)))
24		1cnd 11118	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℂ)
25		nn0cn 12402	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
26		peano2cn 11296	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
28	24, 27	addcomd 11326	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1))
29	25, 24, 24	addassd 11145	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1)))
30		1p1e2 12256	. . . . . . 7 ⊢ (1 + 1) = 2
31	30	oveq2i 7366	. . . . . 6 ⊢ (𝑛 + (1 + 1)) = (𝑛 + 2)
32	31	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2))
33	28, 29, 32	3eqtrd 2772	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2))
34	17, 23, 33	3eqtrd 2772	. . 3 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2))
35	34	mpteq2ia 5190	. 2 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
36	2, 5, 35	3eqtri 2760	1 ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355  ℂcc 11015  0cc0 11017  1c1 11018   + caddc 11020   · cmul 11022  2c2 12191  ℕ₀cn0 12392  IterCompcitco 48819  Ackcack 48820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9542  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-n0 12393  df-z 12480  df-uz 12743  df-seq 13916  df-itco 48821  df-ack 48822

This theorem is referenced by:  ackval2  48844  ackval1012  48852