ackval2 - Mathbox for Alexander van der Vekens

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Theorem ackval2 47321

Description: The Ackermann function at 2. (Contributed by AV, 4-May-2024.)

**Assertion**
Ref	Expression
ackval2	⊢ (Ack‘2) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))

Proof of Theorem ackval2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-2 12271	. . 3 ⊢ 2 = (1 + 1)
2	1	fveq2i 6891	. 2 ⊢ (Ack‘2) = (Ack‘(1 + 1))
3		1nn0 12484	. . 3 ⊢ 1 ∈ ℕ₀
4		ackvalsuc1mpt 47317	. . 3 ⊢ (1 ∈ ℕ₀ → (Ack‘(1 + 1)) = (𝑛 ∈ ℕ₀ ↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)))
5	3, 4	ax-mp 5	. 2 ⊢ (Ack‘(1 + 1)) = (𝑛 ∈ ℕ₀ ↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1))
6		peano2nn0 12508	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
7		2nn0 12485	. . . . . 6 ⊢ 2 ∈ ℕ₀
8		ackval1 47320	. . . . . . 7 ⊢ (Ack‘1) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + 2))
9	8	itcovalpc 47311	. . . . . 6 ⊢ (((𝑛 + 1) ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
10	6, 7, 9	sylancl 586	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
11	10	fveq1d 6890	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1))
12		eqidd 2733	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
13		oveq1 7412	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
14	13	adantl 482	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑖 = 1) → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
15	3	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 1 ∈ ℕ₀)
16		ovexd 7440	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (1 + (2 · (𝑛 + 1))) ∈ V)
17	12, 14, 15, 16	fvmptd 7002	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1) = (1 + (2 · (𝑛 + 1))))
18		nn0cn 12478	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
19		1cnd 11205	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 1 ∈ ℂ)
20		2cnd 12286	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 2 ∈ ℂ)
21		peano2cn 11382	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
22	20, 21	mulcld 11230	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) ∈ ℂ)
23	19, 22	addcomd 11412	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (1 + (2 · (𝑛 + 1))) = ((2 · (𝑛 + 1)) + 1))
24		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25	20, 24, 19	adddid 11234	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
26	25	oveq1d 7420	. . . . . 6 ⊢ (𝑛 ∈ ℂ → ((2 · (𝑛 + 1)) + 1) = (((2 · 𝑛) + (2 · 1)) + 1))
27	20, 24	mulcld 11230	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 𝑛) ∈ ℂ)
28	20, 19	mulcld 11230	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 1) ∈ ℂ)
29	27, 28, 19	addassd 11232	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + ((2 · 1) + 1)))
30		2t1e2 12371	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
31	30	oveq1i 7415	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
32		2p1e3 12350	. . . . . . . . . 10 ⊢ (2 + 1) = 3
33	31, 32	eqtri 2760	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
34	33	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → ((2 · 1) + 1) = 3)
35	34	oveq2d 7421	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) + ((2 · 1) + 1)) = ((2 · 𝑛) + 3))
36	29, 35	eqtrd 2772	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + 3))
37	23, 26, 36	3eqtrd 2776	. . . . 5 ⊢ (𝑛 ∈ ℂ → (1 + (2 · (𝑛 + 1))) = ((2 · 𝑛) + 3))
38	18, 37	syl 17	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (1 + (2 · (𝑛 + 1))) = ((2 · 𝑛) + 3))
39	11, 17, 38	3eqtrd 2776	. . 3 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((2 · 𝑛) + 3))
40	39	mpteq2ia 5250	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))
41	2, 5, 40	3eqtri 2764	1 ⊢ (Ack‘2) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 2c2 12263 3c3 12264 ℕ₀cn0 12468 IterCompcitco 47296 Ackcack 47297

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-itco 47298 df-ack 47299

This theorem is referenced by: ackval3 47322 ackval2012 47330

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