ackval42 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ackval42		Structured version Visualization version GIF version

Theorem ackval42 49053

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 12222	. . . 4 ⊢ 4 = (3 + 1)
2	1	fveq2i 6845	. . 3 ⊢ (Ack‘4) = (Ack‘(3 + 1))
3		df-2 12220	. . 3 ⊢ 2 = (1 + 1)
4	2, 3	fveq12i 6848	. 2 ⊢ ((Ack‘4)‘2) = ((Ack‘(3 + 1))‘(1 + 1))
5		3nn0 12431	. . 3 ⊢ 3 ∈ ℕ₀
6		1nn0 12429	. . 3 ⊢ 1 ∈ ℕ₀
7		ackvalsucsucval 49045	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1)))
8	5, 6, 7	mp2an 693	. 2 ⊢ ((Ack‘(3 + 1))‘(1 + 1)) = ((Ack‘3)‘((Ack‘(3 + 1))‘1))
9		3p1e4 12297	. . . . . . 7 ⊢ (3 + 1) = 4
10	9	fveq2i 6845	. . . . . 6 ⊢ (Ack‘(3 + 1)) = (Ack‘4)
11	10	fveq1i 6843	. . . . 5 ⊢ ((Ack‘(3 + 1))‘1) = ((Ack‘4)‘1)
12		ackval41a 49051	. . . . 5 ⊢ ((Ack‘4)‘1) = ((2↑;16) − 3)
13	11, 12	eqtri 2760	. . . 4 ⊢ ((Ack‘(3 + 1))‘1) = ((2↑;16) − 3)
14	13	fveq2i 6845	. . 3 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((Ack‘3)‘((2↑;16) − 3))
15		2cn 12232	. . . . 5 ⊢ 2 ∈ ℂ
16		6nn0 12434	. . . . . 6 ⊢ 6 ∈ ℕ₀
17	6, 16	deccl 12634	. . . . 5 ⊢ ;16 ∈ ℕ₀
18		expcl 14014	. . . . 5 ⊢ ((2 ∈ ℂ ∧ ;16 ∈ ℕ₀) → (2↑;16) ∈ ℂ)
19	15, 17, 18	mp2an 693	. . . 4 ⊢ (2↑;16) ∈ ℂ
20		3cn 12238	. . . 4 ⊢ 3 ∈ ℂ
21		ackval3 49040	. . . . 5 ⊢ (Ack‘3) = (𝑛 ∈ ℕ₀ ↦ ((2↑(𝑛 + 3)) − 3))
22		oveq1 7375	. . . . . . . 8 ⊢ (𝑛 = ((2↑;16) − 3) → (𝑛 + 3) = (((2↑;16) − 3) + 3))
23		npcan 11401	. . . . . . . 8 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → (((2↑;16) − 3) + 3) = (2↑;16))
24	22, 23	sylan9eqr 2794	. . . . . . 7 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (𝑛 + 3) = (2↑;16))
25	24	oveq2d 7384	. . . . . 6 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → (2↑(𝑛 + 3)) = (2↑(2↑;16)))
26	25	oveq1d 7383	. . . . 5 ⊢ ((((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) ∧ 𝑛 = ((2↑;16) − 3)) → ((2↑(𝑛 + 3)) − 3) = ((2↑(2↑;16)) − 3))
27		3re 12237	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
28		4re 12241	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
29		3lt4 12326	. . . . . . . . . 10 ⊢ 3 < 4
30	27, 28, 29	ltleii 11268	. . . . . . . . 9 ⊢ 3 ≤ 4
31		sq2 14132	. . . . . . . . 9 ⊢ (2↑2) = 4
32	30, 31	breqtrri 5127	. . . . . . . 8 ⊢ 3 ≤ (2↑2)
33		2re 12231	. . . . . . . . 9 ⊢ 2 ∈ ℝ
34		1le2 12361	. . . . . . . . 9 ⊢ 1 ≤ 2
35	17	nn0zi 12528	. . . . . . . . . 10 ⊢ ;16 ∈ ℤ
36		1nn 12168	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
37		2nn0 12430	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
38		9re 12256	. . . . . . . . . . . 12 ⊢ 9 ∈ ℝ
39		2lt9 12357	. . . . . . . . . . . 12 ⊢ 2 < 9
40	33, 38, 39	ltleii 11268	. . . . . . . . . . 11 ⊢ 2 ≤ 9
41	36, 16, 37, 40	declei 12655	. . . . . . . . . 10 ⊢ 2 ≤ ;16
42		2z 12535	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
43	42	eluz1i 12771	. . . . . . . . . 10 ⊢ (;16 ∈ (ℤ_≥‘2) ↔ (;16 ∈ ℤ ∧ 2 ≤ ;16))
44	35, 41, 43	mpbir2an 712	. . . . . . . . 9 ⊢ ;16 ∈ (ℤ_≥‘2)
45		leexp2a 14107	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 1 ≤ 2 ∧ ;16 ∈ (ℤ_≥‘2)) → (2↑2) ≤ (2↑;16))
46	33, 34, 44, 45	mp3an 1464	. . . . . . . 8 ⊢ (2↑2) ≤ (2↑;16)
47		4nn0 12432	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
48	31, 47	eqeltri 2833	. . . . . . . . . 10 ⊢ (2↑2) ∈ ℕ₀
49	48	nn0rei 12424	. . . . . . . . 9 ⊢ (2↑2) ∈ ℝ
50	37, 17	nn0expcli 14023	. . . . . . . . . 10 ⊢ (2↑;16) ∈ ℕ₀
51	50	nn0rei 12424	. . . . . . . . 9 ⊢ (2↑;16) ∈ ℝ
52	27, 49, 51	letri 11274	. . . . . . . 8 ⊢ ((3 ≤ (2↑2) ∧ (2↑2) ≤ (2↑;16)) → 3 ≤ (2↑;16))
53	32, 46, 52	mp2an 693	. . . . . . 7 ⊢ 3 ≤ (2↑;16)
54		nn0sub 12463	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ (2↑;16) ∈ ℕ₀) → (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀))
55	5, 50, 54	mp2an 693	. . . . . . 7 ⊢ (3 ≤ (2↑;16) ↔ ((2↑;16) − 3) ∈ ℕ₀)
56	53, 55	mpbi 230	. . . . . 6 ⊢ ((2↑;16) − 3) ∈ ℕ₀
57	56	a1i 11	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑;16) − 3) ∈ ℕ₀)
58		ovexd 7403	. . . . 5 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((2↑(2↑;16)) − 3) ∈ V)
59	21, 26, 57, 58	fvmptd2 6958	. . . 4 ⊢ (((2↑;16) ∈ ℂ ∧ 3 ∈ ℂ) → ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3))
60	19, 20, 59	mp2an 693	. . 3 ⊢ ((Ack‘3)‘((2↑;16) − 3)) = ((2↑(2↑;16)) − 3)
61		2exp16 17030	. . . . 5 ⊢ (2↑;16) = ;;;;65536
62	61	oveq2i 7379	. . . 4 ⊢ (2↑(2↑;16)) = (2↑;;;;65536)
63	62	oveq1i 7378	. . 3 ⊢ ((2↑(2↑;16)) − 3) = ((2↑;;;;65536) − 3)
64	14, 60, 63	3eqtri 2764	. 2 ⊢ ((Ack‘3)‘((Ack‘(3 + 1))‘1)) = ((2↑;;;;65536) − 3)
65	4, 8, 64	3eqtri 2764	1 ⊢ ((Ack‘4)‘2) = ((2↑;;;;65536) − 3)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 ≤ cle 11179 − cmin 11376 2c2 12212 3c3 12213 4c4 12214 5c5 12215 6c6 12216 9c9 12219 ℕ₀cn0 12413 ℤcz 12500 cdc 12619 ℤ_≥cuz 12763 ↑cexp 13996 Ackcack 49015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-ot 4591 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-rp 12918 df-seq 13937 df-exp 13997 df-itco 49016 df-ack 49017

This theorem is referenced by: ackval42a 49054

W3C validator