Theorem 2exp16 12981

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 9402	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9408	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9212	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9197	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9708	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8169	. 2 ⊢ (2 · 8) = ;16
7		2exp8 12979	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9405	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9608	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9406	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9608	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2229	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9401	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9608	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9403	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9608	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2229	. . . 4 ⊢ ;25 = ;25
18		eqid 2229	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9608	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9608	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9404	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9608	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2229	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2229	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9400	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9615	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2229	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9240	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9261	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9647	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9262	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9653	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9206	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9676	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8307	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9648	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2229	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9263	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9653	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9281	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9243	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6022	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9270	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2250	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9693	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8305	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9653	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9645	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9697	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9201	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9268	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8307	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9653	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9645	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9615	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8305	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2250	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8305	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6021	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8169	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9653	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2250	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9696	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9274	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9653	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9645	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9700	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9653	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9645	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9646	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9208	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8169	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9653	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8169	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9653	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9651	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9701	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9658	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9659	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12969	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1395  (class class class)co 6010  0cc0 8015  1c1 8016   + caddc 8018   · cmul 8020  2c2 9177  3c3 9178  4c4 9179  5c5 9180  6c6 9181  8c8 9183  ;cdc 9594  ↑cexp 10777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-seqfrec 10687  df-exp 10778

This theorem is referenced by: (None)