ackval42 - Mathbox for Alexander van der Vekens

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Theorem ackval42 49626

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 12329	. . . 4 ⊢ 4 = (3 + 1)
2	1	fveq2i 6881	. . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1))
3		df-2 12327	. . 3 ⊢ 2 = (1 + 1)
4	2, 3	fveq12i 6884	. 2 ⊢ ((Ack‘4)‘2) = ((Ack‘(3 + 1))‘(1 + 1))
5		3nn0 12546	. . 3 ⊢ 3 ∈ ℕ₀
6		1nn0 12544	. . 3 ⊢ 1 ∈ ℕ₀
7		ackvalsucsucval 49618	. . 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 12409	. . . . . . 7 ⊢ (3 + 1) = 4
10	9	fveq2i 6881	. . . . . 6 ⊢ (Ack‘(3 + 1)) = (Ack‘4)
11	10	fveq1i 6879	. . . . 5 ⊢ ((Ack‘(3 + 1))‘1) = ((Ack‘4)‘1)
12		ackval41a 49624	. . . . 5 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
13	11, 12	eqtri 2783	. . . 4 ⊢ ((Ack‘(3 + 1))‘1) = ((2↑;16) − 3)
14	13	fveq2i 6881	. . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((Ack‘3)‘((2↑;16) − 3))
15		2cn 12340	. . . . 5 ⊢ 2 ∈ ℂ
16		6nn0 12549	. . . . . 6 ⊢ 6 ∈ ℕ₀
17	6, 16	deccl 12751	. . . . 5 ⊢ ;16 ∈ ℕ₀
18		expcl 14143	. . . . 5 ⊢ ((2 ∈ ℂ ∧ ;16 ∈ ℕ₀) → (2↑;16) ∈ ℂ)
19	15, 17, 18	mp2an 705	. . . 4 ⊢ (2↑;16) ∈ ℂ
20		3cn 12346	. . . 4 ⊢ 3 ∈ ℂ
21		ackval3 49613	. . . . 5 ⊢ (Ack‘3) = (𝑛 ∈ ℕ₀ ↦ ((2↑(𝑛 + 3)) − 3))
22		oveq1 7420	. . . . . . . 8 ⊢ (𝑛 = ((2↑;16) − 3) → (𝑛 + 3) = (((2↑;16) − 3) + 3))
23		npcan 11490	. . . . . . . 8 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → (((2↑;16) − 3) + 3) = (2↑;16))
24	22, 23	sylan9eqr 2817	. . . . . . 7 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (𝑛 + 3) = (2↑;16))
25	24	oveq2d 7429	. . . . . 6 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (2↑(𝑛 + 3)) = (2↑(2↑;16)))
26	25	oveq1d 7428	. . . . 5 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → ((2↑(𝑛 + 3)) − 3) = ((2↑(2↑;16)) − 3))
27		3re 12345	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
28		4re 12349	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
29		3lt4 12441	. . . . . . . . . 10 ⊢ 3 < 4
30	27, 28, 29	ltleii 11357	. . . . . . . . 9 ⊢ 3 ≤ 4
31		sq2 14261	. . . . . . . . 9 ⊢ (2↑2) = 4
32	30, 31	breqtrri 5132	. . . . . . . 8 ⊢ 3 ≤ (2↑2)
33		2re 12339	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34		1le2 12476	. . . . . . . . 9 ⊢ 1 ≤ 2
35	17	nn0zi 12643	. . . . . . . . . 10 ⊢ ;16 ∈ ℤ
36		1nn 12268	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
37		2nn0 12545	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
38		9re 12364	. . . . . . . . . . . 12 ⊢ 9 ∈ ℝ
39		2lt9 12472	. . . . . . . . . . . 12 ⊢ 2 < 9
40	33, 38, 39	ltleii 11357	. . . . . . . . . . 11 ⊢ 2 ≤ 9
41	36, 16, 37, 40	declei 12777	. . . . . . . . . 10 ⊢ 2 ≤ ;16
42		2z 12650	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
43	42	eluz1i 12895	. . . . . . . . . 10 ⊢ (;16 ∈ (ℤ_≥‘2) ↔ (;16 ∈ ℤ ∧ 2 ≤ ;16))
44	35, 41, 43	mpbir2an 724	. . . . . . . . 9 ⊢ ;16 ∈ (ℤ_≥‘2)
45		leexp2a 14236	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ;16 ∈ (ℤ_≥‘2)) → (2↑2) ≤ (2↑;16))
46	33, 34, 44, 45	mp3an 1490	. . . . . . . 8 ⊢ (2↑2) ≤ (2↑;16)
47		4nn0 12547	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
48	31, 47	eqeltri 2856	. . . . . . . . . 10 ⊢ (2↑2) ∈ ℕ₀
49	48	nn0rei 12539	. . . . . . . . 9 ⊢ (2↑2) ∈ ℝ
50	37, 17	nn0expcli 14152	. . . . . . . . . 10 ⊢ (2↑;16) ∈ ℕ₀
51	50	nn0rei 12539	. . . . . . . . 9 ⊢ (2↑;16) ∈ ℝ
52	27, 49, 51	letri 11363	. . . . . . . 8 ⊢ ((3 ≤ (2↑2) ∧ (2↑2) ≤ (2↑;16)) → 3 ≤ (2↑;16))
53	32, 46, 52	mp2an 705	. . . . . . 7 ⊢ 3 ≤ (2↑;16)
54		nn0sub 12578	. . . . . . . 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 7448	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑(2↑;16)) − 3) ∈ V)
59	21, 26, 57, 58	fvmptd2 6995	. . . 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 17182	. . . . 5 ⊢ (2↑;16) = ;;;;65536
62	61	oveq2i 7424	. . . 4 ⊢ (2↑(2↑;16)) = (2↑;;;;65536)
63	62	oveq1i 7423	. . 3 ⊢ ((2↑(2↑;16)) − 3) = ((2↑;;;;65536) − 3)
64	14, 60, 63	3eqtri 2787	. 2 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((2↑;;;;65536) − 3)
65	4, 8, 64	3eqtri 2787	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 2145 Vcvv 3450 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 1c1 11125 + caddc 11127 ≤ cle 11268 − cmin 11465 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 9c9 12326 ℕ₀cn0 12528 ℤcz 12615 cdc 12736 ℤ_≥cuz 12887 ↑cexp 14125 Ackcack 49588

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-rp 13043 df-seq 14066 df-exp 14126 df-itco 49589 df-ack 49590

This theorem is used by: ackval42a 49627

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