2exp16 - Metamath Proof Explorer

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Theorem 2exp16 17128

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 12543	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12549	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12363	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12341	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12848	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11270	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17126	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12546	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12748	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12547	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12748	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2737	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12542	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12748	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12544	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12748	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2737	. . . 4 ⊢ ;25 = ;25
18		eqid 2737	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12748	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12748	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12545	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12748	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2737	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2737	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12541	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12755	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2737	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12388	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12409	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12787	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12411	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12793	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12354	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12816	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11453	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12788	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2737	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12412	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12793	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12430	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7443	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12419	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2765	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12833	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11449	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12793	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12785	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12837	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12347	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12417	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11453	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12793	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12785	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12755	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11449	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2765	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11449	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7442	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11270	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12793	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2765	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12836	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12423	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12793	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12785	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12840	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12793	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12785	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12786	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12357	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11270	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12793	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11270	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12793	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12791	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12841	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12798	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12799	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 17116	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 8c8 12327 cdc 12733 ↑cexp 14102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-seq 14043 df-exp 14103

This theorem is referenced by: 1259lem1 17168 fmtno4 47539 ackval41 48616 ackval42 48617 ackval42a 48618

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