Theorem 2exp16 13128

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 9509	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9515	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9319	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9304	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9819	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8277	. 2 ⊢ (2 · 8) = ;16
7		2exp8 13126	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9512	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9719	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9513	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9719	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2232	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9508	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9719	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9510	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9719	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2232	. . . 4 ⊢ ;25 = ;25
18		eqid 2232	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9719	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9719	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9511	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9719	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2232	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2232	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9507	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9726	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2232	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9347	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9368	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9758	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9369	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9764	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9313	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9787	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8414	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9759	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2232	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9370	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9764	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9388	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9350	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6061	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9377	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2253	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9804	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8412	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9764	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9756	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9808	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9308	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9375	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8414	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9764	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9756	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9726	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8412	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2253	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8412	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6060	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8277	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9764	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2253	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9807	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9381	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9764	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9756	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9811	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9764	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9756	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9757	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9315	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8277	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9764	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8277	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9764	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9762	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9812	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9769	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9770	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 13116	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1398  (class class class)co 6049  0cc0 8123  1c1 8124   + caddc 8126   · cmul 8128  2c2 9284  3c3 9285  4c4 9286  5c5 9287  6c6 9288  8c8 9290  ;cdc 9705  ↑cexp 10896

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 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

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 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-seqfrec 10806  df-exp 10897

This theorem is referenced by: (None)