ackval42 - Mathbox for Alexander van der Vekens

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Theorem ackval42 47897

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 12307	. . . 4 ⊢ 4 = (3 + 1)
2	1	fveq2i 6897	. . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1))
3		df-2 12305	. . 3 ⊢ 2 = (1 + 1)
4	2, 3	fveq12i 6900	. 2 ⊢ ((Ack‘4)‘2) = ((Ack‘(3 + 1))‘(1 + 1))
5		3nn0 12520	. . 3 ⊢ 3 ∈ ℕ₀
6		1nn0 12518	. . 3 ⊢ 1 ∈ ℕ₀
7		ackvalsucsucval 47889	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1)))
8	5, 6, 7	mp2an 690	. 2 ⊢ ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1))
9		3p1e4 12387	. . . . . . 7 ⊢ (3 + 1) = 4
10	9	fveq2i 6897	. . . . . 6 ⊢ (Ack‘(3 + 1)) = (Ack‘4)
11	10	fveq1i 6895	. . . . 5 ⊢ ((Ack‘(3 + 1))‘1) = ((Ack‘4)‘1)
12		ackval41a 47895	. . . . 5 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
13	11, 12	eqtri 2753	. . . 4 ⊢ ((Ack‘(3 + 1))‘1) = ((2↑;16) − 3)
14	13	fveq2i 6897	. . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((Ack‘3)‘((2↑;16) − 3))
15		2cn 12317	. . . . 5 ⊢ 2 ∈ ℂ
16		6nn0 12523	. . . . . 6 ⊢ 6 ∈ ℕ₀
17	6, 16	deccl 12722	. . . . 5 ⊢ ;16 ∈ ℕ₀
18		expcl 14077	. . . . 5 ⊢ ((2 ∈ ℂ ∧ ;16 ∈ ℕ₀) → (2↑;16) ∈ ℂ)
19	15, 17, 18	mp2an 690	. . . 4 ⊢ (2↑;16) ∈ ℂ
20		3cn 12323	. . . 4 ⊢ 3 ∈ ℂ
21		ackval3 47884	. . . . 5 ⊢ (Ack‘3) = (𝑛 ∈ ℕ₀ ↦ ((2↑(𝑛 + 3)) − 3))
22		oveq1 7424	. . . . . . . 8 ⊢ (𝑛 = ((2↑;16) − 3) → (𝑛 + 3) = (((2↑;16) − 3) + 3))
23		npcan 11499	. . . . . . . 8 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → (((2↑;16) − 3) + 3) = (2↑;16))
24	22, 23	sylan9eqr 2787	. . . . . . 7 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (𝑛 + 3) = (2↑;16))
25	24	oveq2d 7433	. . . . . 6 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (2↑(𝑛 + 3)) = (2↑(2↑;16)))
26	25	oveq1d 7432	. . . . 5 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → ((2↑(𝑛 + 3)) − 3) = ((2↑(2↑;16)) − 3))
27		3re 12322	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
28		4re 12326	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
29		3lt4 12416	. . . . . . . . . 10 ⊢ 3 < 4
30	27, 28, 29	ltleii 11367	. . . . . . . . 9 ⊢ 3 ≤ 4
31		sq2 14193	. . . . . . . . 9 ⊢ (2↑2) = 4
32	30, 31	breqtrri 5175	. . . . . . . 8 ⊢ 3 ≤ (2↑2)
33		2re 12316	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34		1le2 12451	. . . . . . . . 9 ⊢ 1 ≤ 2
35	17	nn0zi 12617	. . . . . . . . . 10 ⊢ ;16 ∈ ℤ
36		1nn 12253	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
37		2nn0 12519	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
38		9re 12341	. . . . . . . . . . . 12 ⊢ 9 ∈ ℝ
39		2lt9 12447	. . . . . . . . . . . 12 ⊢ 2 < 9
40	33, 38, 39	ltleii 11367	. . . . . . . . . . 11 ⊢ 2 ≤ 9
41	36, 16, 37, 40	declei 12743	. . . . . . . . . 10 ⊢ 2 ≤ ;16
42		2z 12624	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
43	42	eluz1i 12860	. . . . . . . . . 10 ⊢ (;16 ∈ (ℤ_≥‘2) ↔ (;16 ∈ ℤ ∧ 2 ≤ ;16))
44	35, 41, 43	mpbir2an 709	. . . . . . . . 9 ⊢ ;16 ∈ (ℤ_≥‘2)
45		leexp2a 14169	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ;16 ∈ (ℤ_≥‘2)) → (2↑2) ≤ (2↑;16))
46	33, 34, 44, 45	mp3an 1457	. . . . . . . 8 ⊢ (2↑2) ≤ (2↑;16)
47		4nn0 12521	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
48	31, 47	eqeltri 2821	. . . . . . . . . 10 ⊢ (2↑2) ∈ ℕ₀
49	48	nn0rei 12513	. . . . . . . . 9 ⊢ (2↑2) ∈ ℝ
50	37, 17	nn0expcli 14086	. . . . . . . . . 10 ⊢ (2↑;16) ∈ ℕ₀
51	50	nn0rei 12513	. . . . . . . . 9 ⊢ (2↑;16) ∈ ℝ
52	27, 49, 51	letri 11373	. . . . . . . 8 ⊢ ((3 ≤ (2↑2) ∧ (2↑2) ≤ (2↑;16)) → 3 ≤ (2↑;16))
53	32, 46, 52	mp2an 690	. . . . . . 7 ⊢ 3 ≤ (2↑;16)
54		nn0sub 12552	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (2↑;16) ∈ ℕ₀) → (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀))
55	5, 50, 54	mp2an 690	. . . . . . 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 7452	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑(2↑;16)) − 3) ∈ V)
59	21, 26, 57, 58	fvmptd2 7010	. . . 4 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3))
60	19, 20, 59	mp2an 690	. . 3 ⊢ ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3)
61		2exp16 17060	. . . . 5 ⊢ (2↑;16) = ;;;;65536
62	61	oveq2i 7428	. . . 4 ⊢ (2↑(2↑;16)) = (2↑;;;;65536)
63	62	oveq1i 7427	. . 3 ⊢ ((2↑(2↑;16)) − 3) = ((2↑;;;;65536) − 3)
64	14, 60, 63	3eqtri 2757	. 2 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((2↑;;;;65536) − 3)
65	4, 8, 64	3eqtri 2757	1 ⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 class class class wbr 5148 ‘cfv 6547 (class class class)co 7417 ℂcc 11136 ℝcr 11137 1c1 11139 + caddc 11141 ≤ cle 11279 − cmin 11474 2c2 12297 3c3 12298 4c4 12299 5c5 12300 6c6 12301 9c9 12304 ℕ₀cn0 12502 ℤcz 12588 cdc 12707 ℤ_≥cuz 12852 ↑cexp 14059 Ackcack 47859

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 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-seq 14000 df-exp 14060 df-itco 47860 df-ack 47861

This theorem is referenced by: ackval42a 47898

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