2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16862

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 12323	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12329	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12143	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12121	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12625	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11057	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16860	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12326	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12525	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12327	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12525	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2737	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12322	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12525	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12324	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12525	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2737	. . . 4 ⊢ ;25 = ;25
18		eqid 2737	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12525	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12525	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12325	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12525	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2737	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2737	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12321	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12532	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2737	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12168	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12189	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12564	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12191	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12570	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12134	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12593	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11240	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12565	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2737	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12192	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12570	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12210	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12171	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7327	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12199	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2765	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12610	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 11236	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12570	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12562	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12614	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12127	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12197	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11240	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12570	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12562	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12532	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 11236	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2765	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 11236	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7326	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11057	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12570	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2765	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12613	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12203	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12570	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12562	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12617	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12570	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12562	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12563	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12137	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11057	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12570	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11057	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12570	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12568	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12618	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12575	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12576	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16850	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 (class class class)co 7315 0cc0 10944 1c1 10945 + caddc 10947 · cmul 10949 2c2 12101 3c3 12102 4c4 12103 5c5 12104 6c6 12105 8c8 12107 cdc 12510 ↑cexp 13855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7628 ax-cnex 11000 ax-resscn 11001 ax-1cn 11002 ax-icn 11003 ax-addcl 11004 ax-addrcl 11005 ax-mulcl 11006 ax-mulrcl 11007 ax-mulcom 11008 ax-addass 11009 ax-mulass 11010 ax-distr 11011 ax-i2m1 11012 ax-1ne0 11013 ax-1rid 11014 ax-rnegex 11015 ax-rrecex 11016 ax-cnre 11017 ax-pre-lttri 11018 ax-pre-lttrn 11019 ax-pre-ltadd 11020 ax-pre-mulgt0 11021

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-iun 4939 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5562 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7272 df-ov 7318 df-oprab 7319 df-mpo 7320 df-om 7758 df-2nd 7877 df-frecs 8144 df-wrecs 8175 df-recs 8249 df-rdg 8288 df-er 8546 df-en 8782 df-dom 8783 df-sdom 8784 df-pnf 11084 df-mnf 11085 df-xr 11086 df-ltxr 11087 df-le 11088 df-sub 11280 df-neg 11281 df-nn 12047 df-2 12109 df-3 12110 df-4 12111 df-5 12112 df-6 12113 df-7 12114 df-8 12115 df-9 12116 df-n0 12307 df-z 12393 df-dec 12511 df-uz 12656 df-seq 13795 df-exp 13856

This theorem is referenced by: 1259lem1 16902 fmtno4 45256 ackval41 46293 ackval42 46294 ackval42a 46295

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