Theorem 2exp16 12633

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 9285	. 2 ⊢ 2 ∈ ℕ0
2		8nn0 9291	. 2 ⊢ 8 ∈ ℕ0
3		8cn 9095	. . 3 ⊢ 8 ∈ ℂ
4		2cn 9080	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 9590	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 8052	. 2 ⊢ (2 · 8) = ;16
7		2exp8 12631	. 2 ⊢ (2↑8) = ;;256
8		5nn0 9288	. . . . 5 ⊢ 5 ∈ ℕ0
9	1, 8	deccl 9490	. . . 4 ⊢ ;25 ∈ ℕ0
10		6nn0 9289	. . . 4 ⊢ 6 ∈ ℕ0
11	9, 10	deccl 9490	. . 3 ⊢ ;;256 ∈ ℕ0
12		eqid 2196	. . 3 ⊢ ;;256 = ;;256
13		1nn0 9284	. . . . 5 ⊢ 1 ∈ ℕ0
14	13, 8	deccl 9490	. . . 4 ⊢ ;15 ∈ ℕ0
15		3nn0 9286	. . . 4 ⊢ 3 ∈ ℕ0
16	14, 15	deccl 9490	. . 3 ⊢ ;;153 ∈ ℕ0
17		eqid 2196	. . . 4 ⊢ ;25 = ;25
18		eqid 2196	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 9490	. . . . 5 ⊢ ;12 ∈ ℕ0
20	19, 2	deccl 9490	. . . 4 ⊢ ;;128 ∈ ℕ0
21		4nn0 9287	. . . . . 6 ⊢ 4 ∈ ℕ0
22	13, 21	deccl 9490	. . . . 5 ⊢ ;14 ∈ ℕ0
23		eqid 2196	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2196	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 9283	. . . . . . . 8 ⊢ 0 ∈ ℕ0
26	13	dec0h 9497	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2196	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 9123	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 9144	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 9529	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 9145	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 9535	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 9089	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 9558	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 8190	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 9530	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2196	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 9146	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 9535	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 9164	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 9126	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 5937	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 9153	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2217	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 9575	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 8188	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 9535	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 9527	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 9579	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 9084	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 9151	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 8190	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 9535	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 9527	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 9497	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 8188	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2217	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 8188	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 5936	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 8052	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 9535	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2217	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 9578	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 9157	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 9535	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 9527	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 9582	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 9535	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 9527	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 9528	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 9091	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 8052	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 9535	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 8052	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 9535	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 9533	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 9583	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 9540	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 9541	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 12621	1 ⊢ (2↑;16) = ;;;;65536

Syntax hints:   = wceq 1364  (class class class)co 5925  0cc0 7898  1c1 7899   + caddc 7901   · cmul 7903  2c2 9060  3c3 9061  4c4 9062  5c5 9063  6c6 9064  8c8 9066  ;cdc 9476  ↑cexp 10649

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-5 9071  df-6 9072  df-7 9073  df-8 9074  df-9 9075  df-n0 9269  df-z 9346  df-dec 9477  df-uz 9621  df-seqfrec 10559  df-exp 10650

This theorem is referenced by: (None)