2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16482

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 11951	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 11957	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 11771	. . 3 ⊢ 8 ∈ ℂ
4		2cn 11749	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12252	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 10688	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16480	. 2 ⊢ (2↑8) = ;;256
8		5nn0 11954	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12152	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 11955	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12152	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2758	. . 3 ⊢ ;;256 = ;;256
13		1nn0 11950	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12152	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 11952	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12152	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2758	. . . 4 ⊢ ;25 = ;25
18		eqid 2758	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12152	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12152	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 11953	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12152	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2758	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2758	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 11949	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12159	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2758	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 11796	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 11817	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12191	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 11819	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12197	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 11762	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12220	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 10870	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12192	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2758	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 11820	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12197	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 11838	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 11799	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7162	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 11827	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2781	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12237	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 10866	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12197	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12189	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12241	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 11755	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 11825	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 10870	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12197	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12189	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12159	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 10866	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2781	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 10866	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7161	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 10688	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12197	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2781	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12240	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 11831	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12197	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12189	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12244	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12197	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12189	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12190	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 11765	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 10688	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12197	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 10688	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12197	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12195	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12245	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12202	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12203	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16470	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 (class class class)co 7150 0cc0 10575 1c1 10576 + caddc 10578 · cmul 10580 2c2 11729 3c3 11730 4c4 11731 5c5 11732 6c6 11733 8c8 11735 cdc 12137 ↑cexp 13479

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 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-er 8299 df-en 8528 df-dom 8529 df-sdom 8530 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-seq 13419 df-exp 13480

This theorem is referenced by: 1259lem1 16522 fmtno4 44437 ackval41 45474 ackval42 45475 ackval42a 45476

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