Theorem 2exp16 12631

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 9283	. 2 |- 2 e. NN0
2		8nn0 9289	. 2 |- 8 e. NN0
3		8cn 9093	. . 3 |- 8 e. CC
4		2cn 9078	. . 3 |- 2 e. CC
5		8t2e16 9588	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8050	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 12629	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9286	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9488	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9287	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9488	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2196	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9282	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9488	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9284	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9488	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2196	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2196	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9488	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9488	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9285	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9488	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2196	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2196	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9281	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9495	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2196	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9121	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9142	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9527	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9143	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9533	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9087	. . . . . . 7 |- 5 e. CC
34		8p5e13 9556	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8188	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9528	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2196	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9144	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9533	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9162	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9124	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 5937	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9151	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2217	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9573	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8186	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9533	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9525	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9577	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9082	. . . . . . 7 |- 3 e. CC
51		3p2e5 9149	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8188	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9533	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9525	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9495	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8186	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2217	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8186	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 5936	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8050	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9533	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2217	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9576	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9155	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9533	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9525	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9580	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9533	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9525	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9526	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9089	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8050	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9533	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8050	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9533	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9531	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9581	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9538	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9539	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 12619	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1364  (class class class)co 5925   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   2c2 9058   3c3 9059   4c4 9060   5c5 9061   6c6 9062   8c8 9064  ;cdc 9474   ^cexp 10647

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-seqfrec 10557  df-exp 10648

This theorem is referenced by: (None)