2exp16 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2exp16		Structured version Visualization version GIF version

Theorem 2exp16 17138

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 12570	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12576	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12390	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12368	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12873	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11299	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17136	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12573	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12773	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12574	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12773	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2740	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12569	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12773	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12571	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12773	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2740	. . . 4 ⊢ ;25 = ;25
18		eqid 2740	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12773	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12773	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12572	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12773	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2740	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2740	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12568	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12780	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2740	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12415	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12436	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12812	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12438	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12818	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12381	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12841	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11482	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12813	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2740	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12439	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12818	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12457	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12418	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7460	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12446	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2768	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12858	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11478	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12818	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12810	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12862	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12374	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12444	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11482	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12818	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12810	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12780	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11478	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2768	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11478	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7459	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11299	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12818	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2768	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12861	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12450	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12818	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12810	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12865	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12818	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12810	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12811	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12384	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11299	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12818	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11299	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12818	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12816	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12866	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12823	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12824	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 17126	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 (class class class)co 7448 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 2c2 12348 3c3 12349 4c4 12350 5c5 12351 6c6 12352 8c8 12354 cdc 12758 ↑cexp 14112

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-seq 14053 df-exp 14113

This theorem is referenced by: 1259lem1 17178 fmtno4 47426 ackval41 48429 ackval42 48430 ackval42a 48431

W3C validator