ackval42 - Mathbox for Alexander van der Vekens

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Theorem ackval42 50139

Description: The Ackermann function at (4,2). (Contributed by AV, 9-May-2024.)

**Assertion**
Ref	Expression
ackval42	⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)

**Proof of Theorem ackval42**
Step	Hyp	Ref	Expression
1		df-4 12447	. . . 4 ⊢ 4 = (3 + 1)
2	1	fveq2i 6919	. . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1))
3		df-2 12445	. . 3 ⊢ 2 = (1 + 1)
4	2, 3	fveq12i 6922	. 2 ⊢ ((Ack‘4)‘2) = ((Ack‘(3 + 1))‘(1 + 1))
5		3nn0 12664	. . 3 ⊢ 3 ∈ ℕ₀
6		1nn0 12662	. . 3 ⊢ 1 ∈ ℕ₀
7		ackvalsucsucval 50131	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1)))
8	5, 6, 7	mp2an 705	. 2 ⊢ ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1))
9		3p1e4 12527	. . . . . . 7 ⊢ (3 + 1) = 4
10	9	fveq2i 6919	. . . . . 6 ⊢ (Ack‘(3 + 1)) = (Ack‘4)
11	10	fveq1i 6917	. . . . 5 ⊢ ((Ack‘(3 + 1))‘1) = ((Ack‘4)‘1)
12		ackval41a 50137	. . . . 5 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
13	11, 12	eqtri 2786	. . . 4 ⊢ ((Ack‘(3 + 1))‘1) = ((2↑;16) − 3)
14	13	fveq2i 6919	. . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((Ack‘3)‘((2↑;16) − 3))
15		2cn 12458	. . . . 5 ⊢ 2 ∈ ℂ
16		6nn0 12667	. . . . . 6 ⊢ 6 ∈ ℕ₀
17	6, 16	deccl 12869	. . . . 5 ⊢ ;16 ∈ ℕ₀
18		expcl 14262	. . . . 5 ⊢ ((2 ∈ ℂ ∧ ;16 ∈ ℕ₀) → (2↑;16) ∈ ℂ)
19	15, 17, 18	mp2an 705	. . . 4 ⊢ (2↑;16) ∈ ℂ
20		3cn 12464	. . . 4 ⊢ 3 ∈ ℂ
21		ackval3 50126	. . . . 5 ⊢ (Ack‘3) = (𝑛 ∈ ℕ₀ ↦ ((2↑(𝑛 + 3)) − 3))
22		oveq1 7458	. . . . . . . 8 ⊢ (𝑛 = ((2↑;16) − 3) → (𝑛 + 3) = (((2↑;16) − 3) + 3))
23		npcan 11605	. . . . . . . 8 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → (((2↑;16) − 3) + 3) = (2↑;16))
24	22, 23	sylan9eqr 2820	. . . . . . 7 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (𝑛 + 3) = (2↑;16))
25	24	oveq2d 7467	. . . . . 6 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (2↑(𝑛 + 3)) = (2↑(2↑;16)))
26	25	oveq1d 7466	. . . . 5 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → ((2↑(𝑛 + 3)) − 3) = ((2↑(2↑;16)) − 3))
27		3re 12463	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
28		4re 12467	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
29		3lt4 12559	. . . . . . . . . 10 ⊢ 3 < 4
30	27, 28, 29	ltleii 11472	. . . . . . . . 9 ⊢ 3 ≤ 4
31		sq2 14380	. . . . . . . . 9 ⊢ (2↑2) = 4
32	30, 31	breqtrri 5138	. . . . . . . 8 ⊢ 3 ≤ (2↑2)
33		2re 12457	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34		1le2 12594	. . . . . . . . 9 ⊢ 1 ≤ 2
35	17	nn0zi 12761	. . . . . . . . . 10 ⊢ ;16 ∈ ℤ
36		1nn 12386	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
37		2nn0 12663	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
38		9re 12482	. . . . . . . . . . . 12 ⊢ 9 ∈ ℝ
39		2lt9 12590	. . . . . . . . . . . 12 ⊢ 2 < 9
40	33, 38, 39	ltleii 11472	. . . . . . . . . . 11 ⊢ 2 ≤ 9
41	36, 16, 37, 40	declei 12895	. . . . . . . . . 10 ⊢ 2 ≤ ;16
42		2z 12768	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
43	42	eluz1i 13013	. . . . . . . . . 10 ⊢ (;16 ∈ (ℤ_≥‘2) ↔ (;16 ∈ ℤ ∧ 2 ≤ ;16))
44	35, 41, 43	mpbir2an 724	. . . . . . . . 9 ⊢ ;16 ∈ (ℤ_≥‘2)
45		leexp2a 14355	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ;16 ∈ (ℤ_≥‘2)) → (2↑2) ≤ (2↑;16))
46	33, 34, 44, 45	mp3an 1491	. . . . . . . 8 ⊢ (2↑2) ≤ (2↑;16)
47		4nn0 12665	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
48	31, 47	eqeltri 2860	. . . . . . . . . 10 ⊢ (2↑2) ∈ ℕ₀
49	48	nn0rei 12657	. . . . . . . . 9 ⊢ (2↑2) ∈ ℝ
50	37, 17	nn0expcli 14271	. . . . . . . . . 10 ⊢ (2↑;16) ∈ ℕ₀
51	50	nn0rei 12657	. . . . . . . . 9 ⊢ (2↑;16) ∈ ℝ
52	27, 49, 51	letri 11478	. . . . . . . 8 ⊢ ((3 ≤ (2↑2) ∧ (2↑2) ≤ (2↑;16)) → 3 ≤ (2↑;16))
53	32, 46, 52	mp2an 705	. . . . . . 7 ⊢ 3 ≤ (2↑;16)
54		nn0sub 12696	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (2↑;16) ∈ ℕ₀) → (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀))
55	5, 50, 54	mp2an 705	. . . . . . 7 ⊢ (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀)
56	53, 55	mpbi 233	. . . . . 6 ⊢ ((2↑;16) − 3) ∈ ℕ₀
57	56	a1i 11	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑;16) − 3) ∈ ℕ₀)
58		ovexd 7486	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑(2↑;16)) − 3) ∈ V)
59	21, 26, 57, 58	fvmptd2 7033	. . . 4 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3))
60	19, 20, 59	mp2an 705	. . 3 ⊢ ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3)
61		2exp16 17312	. . . . 5 ⊢ (2↑;16) = ;;;;65536
62	61	oveq2i 7462	. . . 4 ⊢ (2↑(2↑;16)) = (2↑;;;;65536)
63	62	oveq1i 7461	. . 3 ⊢ ((2↑(2↑;16)) − 3) = ((2↑;;;;65536) − 3)
64	14, 60, 63	3eqtri 2790	. 2 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((2↑;;;;65536) − 3)
65	4, 8, 64	3eqtri 2790	1 ⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 1c1 11240 + caddc 11242 ≤ cle 11383 − cmin 11580 2c2 12437 3c3 12438 4c4 12439 5c5 12440 6c6 12441 9c9 12444 ℕ₀cn0 12646 ℤcz 12733 cdc 12854 ℤ_≥cuz 13005 ↑cexp 14244 Ackcack 50101

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-ot 4864 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-rp 13161 df-seq 14185 df-exp 14245 df-itco 50102 df-ack 50103

This theorem is used by: ackval42a 50140

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