Theorem 2exp16 13003

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 9412	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9418	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9222	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9207	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9718	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8179	. 2 ⊢ (2 · 8) = ;16
7		2exp8 13001	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9415	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9618	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9416	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9618	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2229	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9411	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9618	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9413	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9618	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2229	. . . 4 ⊢ ;25 = ;25
18		eqid 2229	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9618	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9618	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9414	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9618	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2229	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2229	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9410	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9625	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2229	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9250	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9271	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9657	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9272	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9663	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9216	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9686	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8317	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9658	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2229	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9273	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9663	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9291	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9253	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6025	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9280	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2250	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9703	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8315	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9663	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9655	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9707	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9211	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9278	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8317	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9663	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9655	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9625	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8315	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2250	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8315	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6024	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8179	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9663	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2250	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9706	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9284	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9663	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9655	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9710	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9663	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9655	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9656	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9218	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8179	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9663	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8179	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9663	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9661	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9711	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9668	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9669	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12991	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1395  (class class class)co 6013  0cc0 8025  1c1 8026   + caddc 8028   · cmul 8030  2c2 9187  3c3 9188  4c4 9189  5c5 9190  6c6 9191  8c8 9193  ;cdc 9604  ↑cexp 10793

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-z 9473  df-dec 9605  df-uz 9749  df-seqfrec 10703  df-exp 10794

This theorem is referenced by: (None)