Theorem 2exp16 12582

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 9263	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9269	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9073	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9058	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9568	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8031	. 2 ⊢ (2 · 8) = ;16
7		2exp8 12580	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9266	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9468	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9267	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9468	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2196	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9262	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9468	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9264	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9468	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2196	. . . 4 ⊢ ;25 = ;25
18		eqid 2196	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9468	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9468	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9265	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9468	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2196	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2196	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9261	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9475	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2196	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9101	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9122	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9507	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9123	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9513	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9067	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9536	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8169	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9508	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2196	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9124	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9513	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9142	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9104	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 5934	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9131	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2217	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9553	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8167	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9513	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9505	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9557	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9062	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9129	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8169	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9513	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9505	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9475	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8167	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2217	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8167	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 5933	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8031	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9513	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2217	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9556	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9135	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9513	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9505	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9560	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9513	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9505	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9506	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9069	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8031	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9513	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8031	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9513	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9511	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9561	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9518	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9519	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12570	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1364  (class class class)co 5922  0cc0 7877  1c1 7878   + caddc 7880   · cmul 7882  2c2 9038  3c3 9039  4c4 9040  5c5 9041  6c6 9042  8c8 9044  ;cdc 9454  ↑cexp 10615

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-mulrcl 7976  ax-addcom 7977  ax-mulcom 7978  ax-addass 7979  ax-mulass 7980  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-1rid 7984  ax-0id 7985  ax-rnegex 7986  ax-precex 7987  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-apti 7992  ax-pre-ltadd 7993  ax-pre-mulgt0 7994  ax-pre-mulext 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-reap 8599  df-ap 8606  df-div 8697  df-inn 8988  df-2 9046  df-3 9047  df-4 9048  df-5 9049  df-6 9050  df-7 9051  df-8 9052  df-9 9053  df-n0 9247  df-z 9324  df-dec 9455  df-uz 9599  df-seqfrec 10525  df-exp 10616

This theorem is referenced by: (None)