Theorem 2exp16 12946

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 9374	. 2 |- 2 e. NN0
2		8nn0 9380	. 2 |- 8 e. NN0
3		8cn 9184	. . 3 |- 8 e. CC
4		2cn 9169	. . 3 |- 2 e. CC
5		8t2e16 9680	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8141	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 12944	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9377	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9580	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9378	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9580	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2229	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9373	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9580	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9375	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9580	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2229	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2229	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9580	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9580	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9376	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9580	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2229	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2229	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9372	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9587	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2229	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9212	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9233	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9619	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9234	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9625	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9178	. . . . . . 7 |- 5 e. CC
34		8p5e13 9648	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8279	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9620	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2229	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9235	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9625	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9253	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9215	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6006	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9242	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2250	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9665	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8277	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9625	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9617	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9669	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9173	. . . . . . 7 |- 3 e. CC
51		3p2e5 9240	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8279	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9625	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9617	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9587	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8277	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2250	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8277	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6005	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8141	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9625	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2250	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9668	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9246	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9625	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9617	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9672	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9625	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9617	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9618	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9180	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8141	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9625	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8141	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9625	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9623	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9673	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9630	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9631	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 12934	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1395  (class class class)co 5994   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   2c2 9149   3c3 9150   4c4 9151   5c5 9152   6c6 9153   8c8 9155  ;cdc 9566   ^cexp 10747

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-seqfrec 10657  df-exp 10748

This theorem is referenced by: (None)