ackval2 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval2		Structured version Visualization version GIF version

Theorem ackval2 47625

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 12276	. . 3 ⊢ 2 = (1 + 1)
2	1	fveq2i 6887	. 2 ⊢ (Ack‘2) = (Ack‘(1 + 1))
3		1nn0 12489	. . 3 ⊢ 1 ∈ ℕ₀
4		ackvalsuc1mpt 47621	. . 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 12513	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
7		2nn0 12490	. . . . . 6 ⊢ 2 ∈ ℕ₀
8		ackval1 47624	. . . . . . 7 ⊢ (Ack‘1) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + 2))
9	8	itcovalpc 47615	. . . . . 6 ⊢ (((𝑛 + 1) ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
10	6, 7, 9	sylancl 585	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
11	10	fveq1d 6886	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1))
12		eqidd 2727	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))) = (𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1)))))
13		oveq1 7411	. . . . . 6 ⊢ (𝑖 = 1 → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
14	13	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑖 = 1) → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1))))
15	3	a1i 11	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 1 ∈ ℕ₀)
16		ovexd 7439	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (1 + (2 · (𝑛 + 1))) ∈ V)
17	12, 14, 15, 16	fvmptd 6998	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → ((𝑖 ∈ ℕ₀ ↦ (𝑖 + (2 · (𝑛 + 1))))‘1) = (1 + (2 · (𝑛 + 1))))
18		nn0cn 12483	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
19		1cnd 11210	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 1 ∈ ℂ)
20		2cnd 12291	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 2 ∈ ℂ)
21		peano2cn 11387	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
22	20, 21	mulcld 11235	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) ∈ ℂ)
23	19, 22	addcomd 11417	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (1 + (2 · (𝑛 + 1))) = ((2 · (𝑛 + 1)) + 1))
24		id 22	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25	20, 24, 19	adddid 11239	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (2 · (𝑛 + 1)) = ((2 · 𝑛) + (2 · 1)))
26	25	oveq1d 7419	. . . . . 6 ⊢ (𝑛 ∈ ℂ → ((2 · (𝑛 + 1)) + 1) = (((2 · 𝑛) + (2 · 1)) + 1))
27	20, 24	mulcld 11235	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 𝑛) ∈ ℂ)
28	20, 19	mulcld 11235	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → (2 · 1) ∈ ℂ)
29	27, 28, 19	addassd 11237	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + ((2 · 1) + 1)))
30		2t1e2 12376	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
31	30	oveq1i 7414	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
32		2p1e3 12355	. . . . . . . . . 10 ⊢ (2 + 1) = 3
33	31, 32	eqtri 2754	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
34	33	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℂ → ((2 · 1) + 1) = 3)
35	34	oveq2d 7420	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → ((2 · 𝑛) + ((2 · 1) + 1)) = ((2 · 𝑛) + 3))
36	29, 35	eqtrd 2766	. . . . . 6 ⊢ (𝑛 ∈ ℂ → (((2 · 𝑛) + (2 · 1)) + 1) = ((2 · 𝑛) + 3))
37	23, 26, 36	3eqtrd 2770	. . . . 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 2770	. . 3 ⊢ (𝑛 ∈ ℕ₀ → (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((2 · 𝑛) + 3))
40	39	mpteq2ia 5244	. 2 ⊢ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))
41	2, 5, 40	3eqtri 2758	1 ⊢ (Ack‘2) = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 3))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3468 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 1c1 11110 + caddc 11112 · cmul 11114 2c2 12268 3c3 12269 ℕ₀cn0 12473 IterCompcitco 47600 Ackcack 47601

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-seq 13970 df-itco 47602 df-ack 47603

This theorem is referenced by: ackval3 47626 ackval2012 47634

W3C validator