2exp16 - Metamath Proof Explorer

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Theorem 2exp16 15997

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 11509	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 11515	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 11306	. . 3 ⊢ 8 ∈ ℂ
4		2cn 11291	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 11853	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 10247	. 2 ⊢ (2 · 8) = ;16
7		2exp8 15996	. 2 ⊢ (2↑8) = ;;256
8		5nn0 11512	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 11712	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 11513	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 11712	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2771	. . 3 ⊢ ;;256 = ;;256
13		1nn0 11508	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 11712	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 11510	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 11712	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2771	. . . 4 ⊢ ;25 = ;25
18		eqid 2771	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 11712	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 11712	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 11511	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 11712	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2771	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2771	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 11507	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 11722	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2771	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 11332	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 11352	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 11769	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 11353	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 11778	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 11300	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 11814	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 10428	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 11771	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2771	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 11354	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 11778	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 11377	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 11334	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6803	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 11362	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2793	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 11833	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 10424	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 11778	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 11765	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 11840	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 11295	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 11360	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 10428	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 11778	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 11765	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 11722	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 10424	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2793	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 10424	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6802	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 10247	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 11778	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2793	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 11838	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 11366	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 11778	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 11765	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 11843	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 11778	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 11765	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 11767	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 11302	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 10247	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 11778	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 10247	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 11778	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 11776	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 11845	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 11786	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 11788	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 15983	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1631 (class class class)co 6791 0cc0 10136 1c1 10137 + caddc 10139 · cmul 10141 2c2 11270 3c3 11271 4c4 11272 5c5 11273 6c6 11274 8c8 11276 cdc 11693 ↑cexp 13060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-seq 13002 df-exp 13061

This theorem is referenced by: 1259lem1 16038 fmtno4 41985

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