ackval2 - Mathbox for Alexander van der Vekens

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Theorem ackval2 47833

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 12313	. . 3 ⊢ 2 = (1 + 1)
2	1	fveq2i 6905	. 2 ⊢ (Ack‘2) = (Ack‘(1 + 1))
3		1nn0 12526	. . 3 ⊢ 1 ∈ ℕ₀
4		ackvalsuc1mpt 47829	. . 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 12550	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
7		2nn0 12527	. . . . . 6 ⊢ 2 ∈ ℕ₀
8		ackval1 47832	. . . . . . 7 ⊢ (Ack‘1) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + 2))
9	8	itcovalpc 47823	. . . . . 6 ⊢ (((𝑛 + 1) ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
10	6, 7, 9	sylancl 584	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
11	10	fveq1d 6904	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1))
12		eqidd 2729	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
13		oveq1 7433	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
14	13	adantl 480	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑖 = 1) → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
15	3	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 1 ∈ ℕ₀)
16		ovexd 7461	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (1 + (2 · (𝑛 + 1))) ∈ V)
17	12, 14, 15, 16	fvmptd 7017	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1) = (1 + (2 · (𝑛 + 1))))
18		nn0cn 12520	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
19		1cnd 11247	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 1 ∈ ℂ)
20		2cnd 12328	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 2 ∈ ℂ)
21		peano2cn 11424	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
22	20, 21	mulcld 11272	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) ∈ ℂ)
23	19, 22	addcomd 11454	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (1 + (2 · (𝑛 + 1))) = ((2 · (𝑛 + 1)) + 1))
24		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25	20, 24, 19	adddid 11276	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
26	25	oveq1d 7441	. . . . . 6 ⊢ (𝑛 ∈ ℂ → ((2 · (𝑛 + 1)) + 1) = (((2 · 𝑛) + (2 · 1)) + 1))
27	20, 24	mulcld 11272	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 𝑛) ∈ ℂ)
28	20, 19	mulcld 11272	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 1) ∈ ℂ)
29	27, 28, 19	addassd 11274	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + ((2 · 1) + 1)))
30		2t1e2 12413	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
31	30	oveq1i 7436	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
32		2p1e3 12392	. . . . . . . . . 10 ⊢ (2 + 1) = 3
33	31, 32	eqtri 2756	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
34	33	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → ((2 · 1) + 1) = 3)
35	34	oveq2d 7442	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) + ((2 · 1) + 1)) = ((2 · 𝑛) + 3))
36	29, 35	eqtrd 2768	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + 3))
37	23, 26, 36	3eqtrd 2772	. . . . 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 2772	. . 3 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((2 · 𝑛) + 3))
40	39	mpteq2ia 5255	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))
41	2, 5, 40	3eqtri 2760	1 ⊢ (Ack‘2) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3473 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 1c1 11147 + caddc 11149 · cmul 11151 2c2 12305 3c3 12306 ℕ₀cn0 12510 IterCompcitco 47808 Ackcack 47809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14007 df-itco 47810 df-ack 47811

This theorem is referenced by: ackval3 47834 ackval2012 47842

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