Theorem 2exp16 12926

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 9354	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9360	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9164	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9149	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9660	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8121	. 2 ⊢ (2 · 8) = ;16
7		2exp8 12924	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9357	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9560	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9358	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9560	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2209	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9353	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9560	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9355	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9560	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2209	. . . 4 ⊢ ;25 = ;25
18		eqid 2209	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9560	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9560	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9356	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9560	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2209	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2209	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9352	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9567	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2209	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9192	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9213	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9599	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9214	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9605	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9158	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9628	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8259	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9600	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2209	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9215	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9605	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9233	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9195	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 5986	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9222	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2230	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9645	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8257	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9605	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9597	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9649	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9153	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9220	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8259	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9605	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9597	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9567	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8257	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2230	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8257	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 5985	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8121	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9605	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2230	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9648	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9226	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9605	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9597	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9652	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9605	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9597	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9598	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9160	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8121	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9605	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8121	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9605	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9603	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9653	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9610	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9611	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12914	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1375  (class class class)co 5974  0cc0 7967  1c1 7968   + caddc 7970   · cmul 7972  2c2 9129  3c3 9130  4c4 9131  5c5 9132  6c6 9133  8c8 9135  ;cdc 9546  ↑cexp 10727

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-uz 9691  df-seqfrec 10637  df-exp 10728

This theorem is referenced by: (None)