2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16773

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 12233	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12239	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12053	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12031	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12534	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 10968	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16771	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12236	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12434	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12237	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12434	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2739	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12232	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12434	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12234	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12434	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2739	. . . 4 ⊢ ;25 = ;25
18		eqid 2739	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12434	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12434	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12235	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12434	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2739	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2739	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12231	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12441	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2739	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12078	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12099	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12473	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12101	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12479	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12044	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12502	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11150	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12474	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2739	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12102	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12479	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12120	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12081	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7280	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12109	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2767	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12519	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 11146	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12479	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12471	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12523	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12037	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12107	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11150	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12479	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12471	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12441	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 11146	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2767	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 11146	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7279	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 10968	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12479	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2767	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12522	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12113	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12479	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12471	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12526	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12479	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12471	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12472	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12047	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 10968	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12479	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 10968	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12479	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12477	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12527	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12484	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12485	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16761	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 (class class class)co 7268 0cc0 10855 1c1 10856 + caddc 10858 · cmul 10860 2c2 12011 3c3 12012 4c4 12013 5c5 12014 6c6 12015 8c8 12017 cdc 12419 ↑cexp 13763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-seq 13703 df-exp 13764

This theorem is referenced by: 1259lem1 16813 fmtno4 44956 ackval41 45993 ackval42 45994 ackval42a 45995

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