2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16416

Description: Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)

**Assertion**
Ref	Expression
2exp16	⊢ (2↑;16) = ;;;;65536

**Proof of Theorem 2exp16**
Step	Hyp	Ref	Expression
1		2nn0 11902	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 11908	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 11722	. . 3 ⊢ 8 ∈ ℂ
4		2cn 11700	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12201	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 10639	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16415	. 2 ⊢ (2↑8) = ;;256
8		5nn0 11905	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12101	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 11906	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12101	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2798	. . 3 ⊢ ;;256 = ;;256
13		1nn0 11901	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12101	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 11903	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12101	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2798	. . . 4 ⊢ ;25 = ;25
18		eqid 2798	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12101	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12101	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 11904	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12101	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2798	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2798	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 11900	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12108	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2798	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 11747	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 11768	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12140	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 11770	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12146	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 11713	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12169	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 10821	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12141	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2798	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 11771	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12146	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 11789	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 11750	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7147	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 11778	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2821	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12186	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 10817	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12146	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12138	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12190	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 11706	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 11776	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 10821	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12146	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12138	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12108	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 10817	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2821	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 10817	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7146	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 10639	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12146	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2821	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12189	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 11782	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12146	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12138	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12193	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12146	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12138	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12139	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 11716	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 10639	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12146	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 10639	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12146	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12144	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12194	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12151	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12152	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16405	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 8c8 11686 cdc 12086 ↑cexp 13425

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13365 df-exp 13426

This theorem is referenced by: 1259lem1 16456 fmtno4 44069 ackval41 45109 ackval42 45110 ackval42a 45111

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