ackval42 - Mathbox for Alexander van der Vekens

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Theorem ackval42 47382

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 12277	. . . 4 ⊢ 4 = (3 + 1)
2	1	fveq2i 6895	. . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1))
3		df-2 12275	. . 3 ⊢ 2 = (1 + 1)
4	2, 3	fveq12i 6898	. 2 ⊢ ((Ack‘4)‘2) = ((Ack‘(3 + 1))‘(1 + 1))
5		3nn0 12490	. . 3 ⊢ 3 ∈ ℕ₀
6		1nn0 12488	. . 3 ⊢ 1 ∈ ℕ₀
7		ackvalsucsucval 47374	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1)))
8	5, 6, 7	mp2an 691	. 2 ⊢ ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1))
9		3p1e4 12357	. . . . . . 7 ⊢ (3 + 1) = 4
10	9	fveq2i 6895	. . . . . 6 ⊢ (Ack‘(3 + 1)) = (Ack‘4)
11	10	fveq1i 6893	. . . . 5 ⊢ ((Ack‘(3 + 1))‘1) = ((Ack‘4)‘1)
12		ackval41a 47380	. . . . 5 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
13	11, 12	eqtri 2761	. . . 4 ⊢ ((Ack‘(3 + 1))‘1) = ((2↑;16) − 3)
14	13	fveq2i 6895	. . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((Ack‘3)‘((2↑;16) − 3))
15		2cn 12287	. . . . 5 ⊢ 2 ∈ ℂ
16		6nn0 12493	. . . . . 6 ⊢ 6 ∈ ℕ₀
17	6, 16	deccl 12692	. . . . 5 ⊢ ;16 ∈ ℕ₀
18		expcl 14045	. . . . 5 ⊢ ((2 ∈ ℂ ∧ ;16 ∈ ℕ₀) → (2↑;16) ∈ ℂ)
19	15, 17, 18	mp2an 691	. . . 4 ⊢ (2↑;16) ∈ ℂ
20		3cn 12293	. . . 4 ⊢ 3 ∈ ℂ
21		ackval3 47369	. . . . 5 ⊢ (Ack‘3) = (𝑛 ∈ ℕ₀ ↦ ((2↑(𝑛 + 3)) − 3))
22		oveq1 7416	. . . . . . . 8 ⊢ (𝑛 = ((2↑;16) − 3) → (𝑛 + 3) = (((2↑;16) − 3) + 3))
23		npcan 11469	. . . . . . . 8 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → (((2↑;16) − 3) + 3) = (2↑;16))
24	22, 23	sylan9eqr 2795	. . . . . . 7 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (𝑛 + 3) = (2↑;16))
25	24	oveq2d 7425	. . . . . 6 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (2↑(𝑛 + 3)) = (2↑(2↑;16)))
26	25	oveq1d 7424	. . . . 5 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → ((2↑(𝑛 + 3)) − 3) = ((2↑(2↑;16)) − 3))
27		3re 12292	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
28		4re 12296	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
29		3lt4 12386	. . . . . . . . . 10 ⊢ 3 < 4
30	27, 28, 29	ltleii 11337	. . . . . . . . 9 ⊢ 3 ≤ 4
31		sq2 14161	. . . . . . . . 9 ⊢ (2↑2) = 4
32	30, 31	breqtrri 5176	. . . . . . . 8 ⊢ 3 ≤ (2↑2)
33		2re 12286	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34		1le2 12421	. . . . . . . . 9 ⊢ 1 ≤ 2
35	17	nn0zi 12587	. . . . . . . . . 10 ⊢ ;16 ∈ ℤ
36		1nn 12223	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
37		2nn0 12489	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
38		9re 12311	. . . . . . . . . . . 12 ⊢ 9 ∈ ℝ
39		2lt9 12417	. . . . . . . . . . . 12 ⊢ 2 < 9
40	33, 38, 39	ltleii 11337	. . . . . . . . . . 11 ⊢ 2 ≤ 9
41	36, 16, 37, 40	declei 12713	. . . . . . . . . 10 ⊢ 2 ≤ ;16
42		2z 12594	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
43	42	eluz1i 12830	. . . . . . . . . 10 ⊢ (;16 ∈ (ℤ_≥‘2) ↔ (;16 ∈ ℤ ∧ 2 ≤ ;16))
44	35, 41, 43	mpbir2an 710	. . . . . . . . 9 ⊢ ;16 ∈ (ℤ_≥‘2)
45		leexp2a 14137	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ;16 ∈ (ℤ_≥‘2)) → (2↑2) ≤ (2↑;16))
46	33, 34, 44, 45	mp3an 1462	. . . . . . . 8 ⊢ (2↑2) ≤ (2↑;16)
47		4nn0 12491	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
48	31, 47	eqeltri 2830	. . . . . . . . . 10 ⊢ (2↑2) ∈ ℕ₀
49	48	nn0rei 12483	. . . . . . . . 9 ⊢ (2↑2) ∈ ℝ
50	37, 17	nn0expcli 14054	. . . . . . . . . 10 ⊢ (2↑;16) ∈ ℕ₀
51	50	nn0rei 12483	. . . . . . . . 9 ⊢ (2↑;16) ∈ ℝ
52	27, 49, 51	letri 11343	. . . . . . . 8 ⊢ ((3 ≤ (2↑2) ∧ (2↑2) ≤ (2↑;16)) → 3 ≤ (2↑;16))
53	32, 46, 52	mp2an 691	. . . . . . 7 ⊢ 3 ≤ (2↑;16)
54		nn0sub 12522	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (2↑;16) ∈ ℕ₀) → (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀))
55	5, 50, 54	mp2an 691	. . . . . . 7 ⊢ (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀)
56	53, 55	mpbi 229	. . . . . 6 ⊢ ((2↑;16) − 3) ∈ ℕ₀
57	56	a1i 11	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑;16) − 3) ∈ ℕ₀)
58		ovexd 7444	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑(2↑;16)) − 3) ∈ V)
59	21, 26, 57, 58	fvmptd2 7007	. . . 4 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3))
60	19, 20, 59	mp2an 691	. . 3 ⊢ ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3)
61		2exp16 17024	. . . . 5 ⊢ (2↑;16) = ;;;;65536
62	61	oveq2i 7420	. . . 4 ⊢ (2↑(2↑;16)) = (2↑;;;;65536)
63	62	oveq1i 7419	. . 3 ⊢ ((2↑(2↑;16)) − 3) = ((2↑;;;;65536) − 3)
64	14, 60, 63	3eqtri 2765	. 2 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((2↑;;;;65536) − 3)
65	4, 8, 64	3eqtri 2765	1 ⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 1c1 11111 + caddc 11113 ≤ cle 11249 − cmin 11444 2c2 12267 3c3 12268 4c4 12269 5c5 12270 6c6 12271 9c9 12274 ℕ₀cn0 12472 ℤcz 12558 cdc 12677 ℤ_≥cuz 12822 ↑cexp 14027 Ackcack 47344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-seq 13967 df-exp 14028 df-itco 47345 df-ack 47346

This theorem is referenced by: ackval42a 47383

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