Theorem 2exp16 12875

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 9347	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9353	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9157	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9142	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9653	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8114	. 2 ⊢ (2 · 8) = ;16
7		2exp8 12873	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9350	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9553	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9351	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9553	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2207	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9346	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9553	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9348	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9553	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2207	. . . 4 ⊢ ;25 = ;25
18		eqid 2207	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9553	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9553	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9349	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9553	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2207	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2207	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9345	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9560	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2207	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9185	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9206	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9592	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9207	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9598	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9151	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9621	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8252	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9593	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2207	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9208	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9598	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9226	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9188	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 5979	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9215	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2228	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9638	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8250	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9598	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9590	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9642	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9146	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9213	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8252	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9598	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9590	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9560	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8250	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2228	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8250	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 5978	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8114	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9598	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2228	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9641	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9219	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9598	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9590	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9645	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9598	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9590	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9591	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9153	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8114	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9598	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8114	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9598	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9596	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9646	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9603	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9604	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12863	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1373  (class class class)co 5967  0cc0 7960  1c1 7961   + caddc 7963   · cmul 7965  2c2 9122  3c3 9123  4c4 9124  5c5 9125  6c6 9126  8c8 9128  ;cdc 9539  ↑cexp 10720

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-dec 9540  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by: (None)