2exp16 - Metamath Proof Explorer

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Theorem 2exp16 17009

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 12409	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12415	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12233	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12211	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12713	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11132	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17007	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12412	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12613	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12413	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12613	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2733	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12408	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12613	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12410	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12613	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2733	. . . 4 ⊢ ;25 = ;25
18		eqid 2733	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12613	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12613	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12411	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12613	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2733	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2733	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12407	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12620	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2733	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12253	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12274	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12652	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12276	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12658	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12224	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12681	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11316	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12653	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2733	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12277	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12658	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12295	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12256	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7367	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12284	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2756	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12698	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11312	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12658	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12650	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12702	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12217	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12282	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11316	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12658	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12650	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12620	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11312	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2756	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11312	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7366	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11132	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12658	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2756	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12701	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12288	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12658	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12650	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12705	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12658	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12650	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12651	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12227	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11132	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12658	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11132	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12658	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12656	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12706	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12663	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12664	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16997	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 (class class class)co 7355 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 2c2 12191 3c3 12192 4c4 12193 5c5 12194 6c6 12195 8c8 12197 cdc 12598 ↑cexp 13975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-seq 13916 df-exp 13976

This theorem is referenced by: 1259lem1 17049 fmtno4 47714 ackval41 48857 ackval42 48858 ackval42a 48859

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