Theorem 2exp16 13071

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 9462	. 2 |- 2 e. NN0
2		8nn0 9468	. 2 |- 8 e. NN0
3		8cn 9272	. . 3 |- 8 e. CC
4		2cn 9257	. . 3 |- 2 e. CC
5		8t2e16 9768	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8229	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 13069	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9465	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9668	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9466	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9668	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2231	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9461	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9668	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9463	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9668	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2231	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2231	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9668	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9668	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9464	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9668	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2231	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2231	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9460	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9675	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2231	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9300	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9321	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9707	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9322	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9713	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9266	. . . . . . 7 |- 5 e. CC
34		8p5e13 9736	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8367	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9708	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2231	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9323	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9713	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9341	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9303	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6040	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9330	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2252	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9753	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8365	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9713	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9705	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9757	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9261	. . . . . . 7 |- 3 e. CC
51		3p2e5 9328	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8367	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9713	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9705	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9675	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8365	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2252	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8365	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6039	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8229	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9713	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2252	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9756	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9334	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9713	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9705	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9760	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9713	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9705	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9706	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9268	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8229	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9713	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8229	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9713	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9711	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9761	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9718	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9719	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 13059	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1398  (class class class)co 6028   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   2c2 9237   3c3 9238   4c4 9239   5c5 9240   6c6 9241   8c8 9243  ;cdc 9654   ^cexp 10844

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-seqfrec 10754  df-exp 10845

This theorem is referenced by: (None)