Theorem 2exp16 13143

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 9518	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9524	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9328	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9313	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9829	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8286	. 2 ⊢ (2 · 8) = ;16
7		2exp8 13141	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9521	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9729	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9522	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9729	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2234	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9517	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9729	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9519	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9729	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2234	. . . 4 ⊢ ;25 = ;25
18		eqid 2234	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9729	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9729	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9520	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9729	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2234	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2234	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9516	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9736	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2234	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9356	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9377	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9768	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9378	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9774	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9322	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9797	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8423	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9769	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2234	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9379	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9774	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9397	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9359	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6064	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9386	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2255	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9814	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8421	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9774	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9766	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9818	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9317	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9384	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8423	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9774	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9766	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9736	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8421	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2255	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8421	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6063	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8286	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9774	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2255	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9817	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9390	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9774	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9766	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9821	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9774	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9766	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9767	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9324	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8286	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9774	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8286	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9774	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9772	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9822	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9779	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9780	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 13131	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1398  (class class class)co 6052  0cc0 8132  1c1 8133   + caddc 8135   · cmul 8137  2c2 9293  3c3 9294  4c4 9295  5c5 9296  6c6 9297  8c8 9299  ;cdc 9715  ↑cexp 10907

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-dec 9716  df-uz 9860  df-seqfrec 10817  df-exp 10908

This theorem is referenced by: (None)