Theorem 2exp16 13166

Description: Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
2exp16	|- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Proof of Theorem 2exp16

Step	Hyp	Ref	Expression
1		2nn0 9535	. 2 |- 2 e. NN0
2		8nn0 9541	. 2 |- 8 e. NN0
3		8cn 9345	. . 3 |- 8 e. CC
4		2cn 9330	. . 3 |- 2 e. CC
5		8t2e16 9846	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8299	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 13164	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9538	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9746	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9539	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9746	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2234	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9534	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9746	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9536	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9746	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2234	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2234	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9746	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9746	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9537	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9746	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2234	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2234	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9533	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9753	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2234	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9373	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9394	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9785	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9395	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9791	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9339	. . . . . . 7 |- 5 e. CC
34		8p5e13 9814	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8437	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9786	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2234	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9396	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9791	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9414	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9376	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6072	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9403	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2255	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9831	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8435	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9791	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9783	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9835	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9334	. . . . . . 7 |- 3 e. CC
51		3p2e5 9401	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8437	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9791	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9783	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9753	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8435	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2255	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8435	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6071	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8299	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9791	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2255	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9834	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9407	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9791	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9783	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9838	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9791	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9783	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9784	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9341	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8299	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9791	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8299	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9791	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9789	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9839	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9796	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9797	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 13154	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1398  (class class class)co 6060   0cc0 8145   1c1 8146   + caddc 8148   x. cmul 8150   2c2 9310   3c3 9311   4c4 9312   5c5 9313   6c6 9314   8c8 9316  ;cdc 9732   ^cexp 10929

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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-frec 6637  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-5 9321  df-6 9322  df-7 9323  df-8 9324  df-9 9325  df-n0 9519  df-z 9600  df-dec 9733  df-uz 9877  df-seqfrec 10839  df-exp 10930

This theorem is referenced by: (None)