Theorem 2exp16 13239

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 9585	. 2 |- 2 e. NN0
2		8nn0 9591	. 2 |- 8 e. NN0
3		8cn 9393	. . 3 |- 8 e. CC
4		2cn 9378	. . 3 |- 2 e. CC
5		8t2e16 9901	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8334	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 13237	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9588	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9796	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9589	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9796	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2238	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9584	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9796	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9586	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9796	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2238	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2238	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9796	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9796	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9587	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9796	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2238	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2238	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9583	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9808	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2238	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9421	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9442	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9840	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9443	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9846	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9387	. . . . . . 7 |- 5 e. CC
34		8p5e13 9869	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8473	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9841	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2238	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9444	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9846	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9462	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9424	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6097	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9451	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2259	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9886	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8471	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9846	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9838	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9890	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9382	. . . . . . 7 |- 3 e. CC
51		3p2e5 9449	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8473	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9846	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9838	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9808	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8471	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2259	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8471	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6096	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8334	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9846	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2259	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9889	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9455	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9846	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9838	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9893	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9846	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9838	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9839	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9389	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8334	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9846	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8334	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9846	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9844	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9894	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9851	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9852	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 13227	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

This proof depends on syntax axioms:   = wceq 1402  (class class class)co 6085   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   2c2 9358   3c3 9359   4c4 9360   5c5 9361   6c6 9362   8c8 9364  ;cdc 9782   ^cexp 10989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-uz 9932  df-seqfrec 10899  df-exp 10990

This theorem is used by:  1259lem1  13264