Theorem ackval1 48602

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 12775	. . 3 ⊢ 1 = (0 + 1)
2	1	fveq2i 6909	. 2 ⊢ (Ack‘1) = (Ack‘(0 + 1))
3		0nn0 12541	. . 3 ⊢ 0 ∈ ℕ0
4		ackvalsuc1mpt 48599	. . 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 12566	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7		1nn0 12542	. . . . . . 7 ⊢ 1 ∈ ℕ0
8		ackval0 48601	. . . . . . . 8 ⊢ (Ack‘0) = (𝑖 ∈ ℕ0 ↦ (𝑖 + 1))
9	8	itcovalpc 48593	. . . . . . 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 12536	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
13	12	mullidd 11279	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (1 · (𝑛 + 1)) = (𝑛 + 1))
14	13	oveq2d 7447	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → (𝑖 + (1 · (𝑛 + 1))) = (𝑖 + (𝑛 + 1)))
15	14	mpteq2dv 5244	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (1 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
16	10, 15	eqtrd 2777	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((IterComp‘(Ack‘0))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
17	16	fveq1d 6908	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1))
18		eqidd 2738	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1))))
19		oveq1 7438	. . . . . 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 7466	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) ∈ V)
23	18, 20, 21, 22	fvmptd 7023	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑖 ∈ ℕ0 ↦ (𝑖 + (𝑛 + 1)))‘1) = (1 + (𝑛 + 1)))
24		1cnd 11256	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℂ)
25		nn0cn 12536	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
26		peano2cn 11433	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℂ)
28	24, 27	addcomd 11463	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = ((𝑛 + 1) + 1))
29	25, 24, 24	addassd 11283	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 + 1) + 1) = (𝑛 + (1 + 1)))
30		1p1e2 12391	. . . . . . 7 ⊢ (1 + 1) = 2
31	30	oveq2i 7442	. . . . . 6 ⊢ (𝑛 + (1 + 1)) = (𝑛 + 2)
32	31	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + (1 + 1)) = (𝑛 + 2))
33	28, 29, 32	3eqtrd 2781	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (1 + (𝑛 + 1)) = (𝑛 + 2))
34	17, 23, 33	3eqtrd 2781	. . 3 ⊢ (𝑛 ∈ ℕ0 → (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1) = (𝑛 + 2))
35	34	mpteq2ia 5245	. 2 ⊢ (𝑛 ∈ ℕ0 ↦ (((IterComp‘(Ack‘0))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))
36	2, 5, 35	3eqtri 2769	1 ⊢ (Ack‘1) = (𝑛 ∈ ℕ0 ↦ (𝑛 + 2))

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  2c2 12321  ℕ₀cn0 12526  IterCompcitco 48578  Ackcack 48579

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-itco 48580  df-ack 48581

This theorem is referenced by:  ackval2  48603  ackval1012  48611