2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16029

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 11596	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 11602	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 11415	. . 3 ⊢ 8 ∈ ℂ
4		2cn 11388	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 11894	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 10344	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16028	. 2 ⊢ (2↑8) = ;;256
8		5nn0 11599	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 11794	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 11600	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 11794	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2817	. . 3 ⊢ ;;256 = ;;256
13		1nn0 11595	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 11794	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 11597	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 11794	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2817	. . . 4 ⊢ ;25 = ;25
18		eqid 2817	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 11794	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 11794	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 11598	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 11794	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2817	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2817	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 11594	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 11801	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2817	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 11442	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 11463	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 11833	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 11464	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 11839	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 11403	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 11862	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 10523	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 11834	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2817	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 11465	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 11839	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 11483	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 11445	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6896	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 11472	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2839	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 11879	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 10519	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 11839	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 11831	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 11883	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 11394	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 11470	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 10523	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 11839	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 11831	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 11801	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 10519	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2839	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 10519	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6895	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 10344	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 11839	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2839	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 11882	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 11476	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 11839	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 11831	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 11886	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 11839	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 11831	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 11832	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 11407	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 10344	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 11839	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 10344	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 11839	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 11837	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 11887	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 11844	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 11845	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16020	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1637 (class class class)co 6884 0cc0 10231 1c1 10232 + caddc 10234 · cmul 10236 2c2 11368 3c3 11369 4c4 11370 5c5 11371 6c6 11372 8c8 11374 cdc 11779 ↑cexp 13103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-sep 4988 ax-nul 4996 ax-pow 5048 ax-pr 5109 ax-un 7189 ax-cnex 10287 ax-resscn 10288 ax-1cn 10289 ax-icn 10290 ax-addcl 10291 ax-addrcl 10292 ax-mulcl 10293 ax-mulrcl 10294 ax-mulcom 10295 ax-addass 10296 ax-mulass 10297 ax-distr 10298 ax-i2m1 10299 ax-1ne0 10300 ax-1rid 10301 ax-rnegex 10302 ax-rrecex 10303 ax-cnre 10304 ax-pre-lttri 10305 ax-pre-lttrn 10306 ax-pre-ltadd 10307 ax-pre-mulgt0 10308

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-iun 4725 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5232 df-eprel 5237 df-po 5245 df-so 5246 df-fr 5283 df-we 5285 df-xp 5330 df-rel 5331 df-cnv 5332 df-co 5333 df-dm 5334 df-rn 5335 df-res 5336 df-ima 5337 df-pred 5907 df-ord 5953 df-on 5954 df-lim 5955 df-suc 5956 df-iota 6074 df-fun 6113 df-fn 6114 df-f 6115 df-f1 6116 df-fo 6117 df-f1o 6118 df-fv 6119 df-riota 6845 df-ov 6887 df-oprab 6888 df-mpt2 6889 df-om 7306 df-2nd 7409 df-wrecs 7652 df-recs 7714 df-rdg 7752 df-er 7989 df-en 8203 df-dom 8204 df-sdom 8205 df-pnf 10371 df-mnf 10372 df-xr 10373 df-ltxr 10374 df-le 10375 df-sub 10563 df-neg 10564 df-nn 11316 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11580 df-z 11664 df-dec 11780 df-uz 11925 df-seq 13045 df-exp 13104

This theorem is referenced by: 1259lem1 16069 fmtno4 42057

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