Theorem 2exp16 15797

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 11309	. 2 |- 2 e. NN0
2		8nn0 11315	. 2 |- 8 e. NN0
3		8cn 11106	. . 3 |- 8 e. CC
4		2cn 11091	. . 3 |- 2 e. CC
5		8t2e16 11654	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 10047	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 15796	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 11312	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 11512	. . . 4 |- ; 2 5 e. NN0
10		6nn0 11313	. . . 4 |- 6 e. NN0
11	9, 10	deccl 11512	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2622	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 11308	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 11512	. . . 4 |- ; 1 5 e. NN0
15		3nn0 11310	. . . 4 |- 3 e. NN0
16	14, 15	deccl 11512	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2622	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2622	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 11512	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 11512	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 11311	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 11512	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2622	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2622	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 11307	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 11522	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2622	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 11132	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 11152	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 11570	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 11153	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 11579	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 11100	. . . . . . 7 |- 5 e. CC
34		8p5e13 11615	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 10228	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 11572	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2622	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 11154	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 11579	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 11177	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 11134	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6662	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 11162	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2644	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 11634	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addid2i 10224	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 11579	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 11566	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 11641	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 11095	. . . . . . 7 |- 3 e. CC
51		3p2e5 11160	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 10228	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 11579	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 11566	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 11522	. . . . 5 |- 3 = ; 0 3
56	50	addid2i 10224	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2644	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addid2i 10224	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6661	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2644	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 11639	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 11166	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 11579	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 11566	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 11644	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 11579	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 11566	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 11568	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 11102	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 10047	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 11579	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 10047	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 11579	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 11577	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 11646	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 11587	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 11589	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 15783	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1483  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   8c8 11076  ;cdc 11493   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  1259lem1  15838  fmtno4  41464