2exp16 - Metamath Proof Explorer

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Theorem 2exp16 17024

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 12489	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12495	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12309	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12287	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12792	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11223	. 2 ⊢ (2 · 8) = ;16
7		2exp8 17022	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12492	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12692	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12493	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12692	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2733	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12488	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12692	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12490	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12692	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2733	. . . 4 ⊢ ;25 = ;25
18		eqid 2733	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12692	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12692	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12491	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12692	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2733	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2733	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12487	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12699	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2733	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12334	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12355	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12731	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12357	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12737	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12300	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12760	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11406	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12732	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2733	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12358	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12737	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12376	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12337	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7421	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12365	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2761	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12777	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addlidi 11402	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12737	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12729	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12781	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12293	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12363	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11406	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12737	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12729	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12699	. . . . 5 ⊢ 3 = ;03
56	50	addlidi 11402	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2761	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addlidi 11402	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7420	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11223	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12737	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2761	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12780	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12369	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12737	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12729	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12784	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12737	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12729	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12730	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12303	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11223	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12737	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11223	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12737	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12735	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12785	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12742	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12743	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 17012	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 2c2 12267 3c3 12268 4c4 12269 5c5 12270 6c6 12271 8c8 12273 cdc 12677 ↑cexp 14027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-seq 13967 df-exp 14028

This theorem is referenced by: 1259lem1 17064 fmtno4 46268 ackval41 47429 ackval42 47430 ackval42a 47431

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