Theorem 2exp16 13131

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 9512	. 2 |- 2 e. NN0
2		8nn0 9518	. 2 |- 8 e. NN0
3		8cn 9322	. . 3 |- 8 e. CC
4		2cn 9307	. . 3 |- 2 e. CC
5		8t2e16 9822	. . 3 |- ( 8 x. 2 ) = ; 1 6
6	3, 4, 5	mulcomli 8280	. 2 |- ( 2 x. 8 ) = ; 1 6
7		2exp8 13129	. 2 |- ( 2 ^ 8 ) = ;; 2 5 6
8		5nn0 9515	. . . . 5 |- 5 e. NN0
9	1, 8	deccl 9722	. . . 4 |- ; 2 5 e. NN0
10		6nn0 9516	. . . 4 |- 6 e. NN0
11	9, 10	deccl 9722	. . 3 |- ;; 2 5 6 e. NN0
12		eqid 2232	. . 3 |- ;; 2 5 6 = ;; 2 5 6
13		1nn0 9511	. . . . 5 |- 1 e. NN0
14	13, 8	deccl 9722	. . . 4 |- ; 1 5 e. NN0
15		3nn0 9513	. . . 4 |- 3 e. NN0
16	14, 15	deccl 9722	. . 3 |- ;; 1 5 3 e. NN0
17		eqid 2232	. . . 4 |- ; 2 5 = ; 2 5
18		eqid 2232	. . . 4 |- ;; 1 5 3 = ;; 1 5 3
19	13, 1	deccl 9722	. . . . 5 |- ; 1 2 e. NN0
20	19, 2	deccl 9722	. . . 4 |- ;; 1 2 8 e. NN0
21		4nn0 9514	. . . . . 6 |- 4 e. NN0
22	13, 21	deccl 9722	. . . . 5 |- ; 1 4 e. NN0
23		eqid 2232	. . . . . 6 |- ; 1 5 = ; 1 5
24		eqid 2232	. . . . . 6 |- ;; 1 2 8 = ;; 1 2 8
25		0nn0 9510	. . . . . . . 8 |- 0 e. NN0
26	13	dec0h 9729	. . . . . . . 8 |- 1 = ; 0 1
27		eqid 2232	. . . . . . . 8 |- ; 1 2 = ; 1 2
28		0p1e1 9350	. . . . . . . 8 |- ( 0 + 1 ) = 1
29		1p2e3 9371	. . . . . . . 8 |- ( 1 + 2 ) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9761	. . . . . . 7 |- ( 1 + ; 1 2 ) = ; 1 3
31		3p1e4 9372	. . . . . . 7 |- ( 3 + 1 ) = 4
32	13, 15, 13, 30, 31	decaddi 9767	. . . . . 6 |- ( ( 1 + ; 1 2 ) + 1 ) = ; 1 4
33		5cn 9316	. . . . . . 7 |- 5 e. CC
34		8p5e13 9790	. . . . . . 7 |- ( 8 + 5 ) = ; 1 3
35	3, 33, 34	addcomli 8417	. . . . . 6 |- ( 5 + 8 ) = ; 1 3
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9762	. . . . 5 |- (; 1 5 + ;; 1 2 8 ) = ;; 1 4 3
37		eqid 2232	. . . . . . 7 |- ; 1 4 = ; 1 4
38		4p1e5 9373	. . . . . . 7 |- ( 4 + 1 ) = 5
39	13, 21, 13, 37, 38	decaddi 9767	. . . . . 6 |- (; 1 4 + 1 ) = ; 1 5
40		2t2e4 9391	. . . . . . . 8 |- ( 2 x. 2 ) = 4
41		1p1e2 9353	. . . . . . . 8 |- ( 1 + 1 ) = 2
42	40, 41	oveq12i 6061	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = ( 4 + 2 )
43		4p2e6 9380	. . . . . . 7 |- ( 4 + 2 ) = 6
44	42, 43	eqtri 2253	. . . . . 6 |- ( ( 2 x. 2 ) + ( 1 + 1 ) ) = 6
45		5t2e10 9807	. . . . . . 7 |- ( 5 x. 2 ) = ; 1 0
46	33	addlidi 8415	. . . . . . 7 |- ( 0 + 5 ) = 5
47	13, 25, 8, 45, 46	decaddi 9767	. . . . . 6 |- ( ( 5 x. 2 ) + 5 ) = ; 1 5
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9759	. . . . 5 |- ( (; 2 5 x. 2 ) + (; 1 4 + 1 ) ) = ; 6 5
49		6t2e12 9811	. . . . . 6 |- ( 6 x. 2 ) = ; 1 2
50		3cn 9311	. . . . . . 7 |- 3 e. CC
51		3p2e5 9378	. . . . . . 7 |- ( 3 + 2 ) = 5
52	50, 4, 51	addcomli 8417	. . . . . 6 |- ( 2 + 3 ) = 5
53	13, 1, 15, 49, 52	decaddi 9767	. . . . 5 |- ( ( 6 x. 2 ) + 3 ) = ; 1 5
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9759	. . . 4 |- ( (;; 2 5 6 x. 2 ) + (; 1 5 + ;; 1 2 8 ) ) = ;; 6 5 5
55	15	dec0h 9729	. . . . 5 |- 3 = ; 0 3
56	50	addlidi 8415	. . . . . . 7 |- ( 0 + 3 ) = 3
57	56, 55	eqtri 2253	. . . . . 6 |- ( 0 + 3 ) = ; 0 3
58	4	addlidi 8415	. . . . . . . 8 |- ( 0 + 2 ) = 2
59	58	oveq2i 6060	. . . . . . 7 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ( ( 2 x. 5 ) + 2 )
60	33, 4, 45	mulcomli 8280	. . . . . . . 8 |- ( 2 x. 5 ) = ; 1 0
61	13, 25, 1, 60, 58	decaddi 9767	. . . . . . 7 |- ( ( 2 x. 5 ) + 2 ) = ; 1 2
62	59, 61	eqtri 2253	. . . . . 6 |- ( ( 2 x. 5 ) + ( 0 + 2 ) ) = ; 1 2
63		5t5e25 9810	. . . . . . 7 |- ( 5 x. 5 ) = ; 2 5
64		5p3e8 9384	. . . . . . 7 |- ( 5 + 3 ) = 8
65	1, 8, 15, 63, 64	decaddi 9767	. . . . . 6 |- ( ( 5 x. 5 ) + 3 ) = ; 2 8
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9759	. . . . 5 |- ( (; 2 5 x. 5 ) + ( 0 + 3 ) ) = ;; 1 2 8
67		6t5e30 9814	. . . . . 6 |- ( 6 x. 5 ) = ; 3 0
68	15, 25, 15, 67, 56	decaddi 9767	. . . . 5 |- ( ( 6 x. 5 ) + 3 ) = ; 3 3
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9759	. . . 4 |- ( (;; 2 5 6 x. 5 ) + 3 ) = ;;; 1 2 8 3
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9760	. . 3 |- ( (;; 2 5 6 x. ; 2 5 ) + ;; 1 5 3 ) = ;;; 6 5 5 3
71		6cn 9318	. . . . . . 7 |- 6 e. CC
72	71, 4, 49	mulcomli 8280	. . . . . 6 |- ( 2 x. 6 ) = ; 1 2
73	13, 1, 15, 72, 52	decaddi 9767	. . . . 5 |- ( ( 2 x. 6 ) + 3 ) = ; 1 5
74	71, 33, 67	mulcomli 8280	. . . . . 6 |- ( 5 x. 6 ) = ; 3 0
75	15, 25, 15, 74, 56	decaddi 9767	. . . . 5 |- ( ( 5 x. 6 ) + 3 ) = ; 3 3
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9765	. . . 4 |- ( (; 2 5 x. 6 ) + 3 ) = ;; 1 5 3
77		6t6e36 9815	. . . 4 |- ( 6 x. 6 ) = ; 3 6
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9772	. . 3 |- (;; 2 5 6 x. 6 ) = ;;; 1 5 3 6
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9773	. 2 |- (;; 2 5 6 x. ;; 2 5 6 ) = ;;;; 6 5 5 3 6
80	1, 2, 6, 7, 79	numexp2x 13119	1 |- ( 2 ^; 1 6 ) = ;;;; 6 5 5 3 6

Syntax hints:   = wceq 1398  (class class class)co 6049   0cc0 8126   1c1 8127   + caddc 8129   x. cmul 8131   2c2 9287   3c3 9288   4c4 9289   5c5 9290   6c6 9291   8c8 9293  ;cdc 9708   ^cexp 10899

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-z 9577  df-dec 9709  df-uz 9853  df-seqfrec 10809  df-exp 10900

This theorem is referenced by: (None)