Theorem 2exp16 13033

Description: Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)

Assertion

Ref	Expression
2exp16	⊢ (2↑;16) = ;;;;65536

Proof of Theorem 2exp16

Step	Hyp	Ref	Expression
1		2nn0 9424	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9430	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9234	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9219	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9730	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8191	. 2 ⊢ (2 · 8) = ;16
7		2exp8 13031	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9427	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9630	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9428	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9630	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2230	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9423	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9630	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9425	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9630	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2230	. . . 4 ⊢ ;25 = ;25
18		eqid 2230	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9630	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9630	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9426	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9630	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2230	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2230	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9422	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9637	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2230	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9262	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9283	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9669	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9284	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9675	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9228	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9698	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8329	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9670	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2230	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9285	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9675	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9303	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9265	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6035	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9292	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2251	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9715	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8327	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9675	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9667	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9719	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9223	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9290	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8329	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9675	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9667	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9637	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8327	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2251	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8327	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6034	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8191	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9675	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2251	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9718	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9296	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9675	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9667	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9722	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9675	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9667	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9668	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9230	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8191	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9675	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8191	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9675	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9673	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9723	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9680	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9681	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 13021	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1397  (class class class)co 6023  0cc0 8037  1c1 8038   + caddc 8040   · cmul 8042  2c2 9199  3c3 9200  4c4 9201  5c5 9202  6c6 9203  8c8 9205  ;cdc 9616  ↑cexp 10806

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-seqfrec 10716  df-exp 10807

This theorem is referenced by: (None)