Theorem 2exp16 13015

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 9419	. 2 |- 2 e. NN0
2		8nn0 9425	. 2 |- 8 e. NN0
3		8cn 9229	. . 3 |- 8 e. CC
4		2cn 9214	. . 3 |- 2 e. CC
5		8t2e16 9725	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8186	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 13013	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9422	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9625	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9423	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9625	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2231	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9418	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9625	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9420	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9625	. . 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 9625	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9625	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9421	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9625	. . . . 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 9417	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9632	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2231	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9257	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9278	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9664	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9279	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9670	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9223	. . . . . . 7 |- 5 e. CC
34		8p5e13 9693	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8324	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9665	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2231	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9280	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9670	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9298	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9260	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6030	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9287	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2252	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9710	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8322	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9670	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9662	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9714	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9218	. . . . . . 7 |- 3 e. CC
51		3p2e5 9285	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8324	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9670	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9662	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9632	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8322	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2252	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8322	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6029	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8186	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9670	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2252	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9713	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9291	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9670	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9662	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9717	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9670	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9662	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9663	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9225	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8186	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9670	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8186	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9670	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9668	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9718	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9675	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9676	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 13003	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1397  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   2c2 9194   3c3 9195   4c4 9196   5c5 9197   6c6 9198   8c8 9200  ;cdc 9611   ^cexp 10801

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-seqfrec 10711  df-exp 10802

This theorem is referenced by: (None)