2exp16 - Metamath Proof Explorer

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Theorem 2exp16 17063

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 12522	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12528	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12342	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12320	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12825	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11255	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17061	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12525	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12725	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12526	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12725	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2725	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12521	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12725	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12523	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12725	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2725	. . . 4 ⊢ ;25 = ;25
18		eqid 2725	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12725	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12725	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12524	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12725	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2725	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2725	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12520	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12732	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2725	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12367	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12388	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12764	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12390	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12770	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12333	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12793	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11438	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12765	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2725	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12391	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12770	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12409	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12370	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7431	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12398	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2753	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12810	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11434	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12770	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12762	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12814	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12326	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12396	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11438	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12770	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12762	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12732	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11434	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2753	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11434	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7430	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11255	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12770	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2753	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12813	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12402	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12770	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12762	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12817	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12770	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12762	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12763	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12336	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11255	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12770	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11255	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12770	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12768	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12818	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12775	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12776	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 17051	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 2c2 12300 3c3 12301 4c4 12302 5c5 12303 6c6 12304 8c8 12306 cdc 12710 ↑cexp 14062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-seq 14003 df-exp 14063

This theorem is referenced by: 1259lem1 17103 fmtno4 47026 ackval41 47951 ackval42 47952 ackval42a 47953

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