2exp16 - Metamath Proof Explorer

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Theorem 2exp16 16841

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 12300	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 12306	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 12120	. . 3 ⊢ 8 ∈ ℂ
4		2cn 12098	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 12602	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 11034	. 2 ⊢ (2 · 8) = ;16
7		2exp8 16839	. 2 ⊢ (2↑8) = ;;256
8		5nn0 12303	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 12502	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 12304	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12502	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2736	. . 3 ⊢ ;;256 = ;;256
13		1nn0 12299	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 12502	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 12301	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 12502	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2736	. . . 4 ⊢ ;25 = ;25
18		eqid 2736	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 12502	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 12502	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 12302	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 12502	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2736	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2736	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 12298	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 12509	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2736	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 12145	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 12166	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 12541	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 12168	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 12547	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 12111	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 12570	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 11217	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 12542	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2736	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 12169	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 12547	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 12187	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 12148	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 7319	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 12176	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2764	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 12587	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 11213	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 12547	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 12539	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 12591	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 12104	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 12174	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 11217	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 12547	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 12539	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 12509	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 11213	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2764	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 11213	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 7318	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 11034	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 12547	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2764	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 12590	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 12180	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 12547	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 12539	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 12594	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 12547	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 12539	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 12540	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 12114	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 11034	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 12547	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 11034	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 12547	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 12545	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 12595	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 12552	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 12553	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 16829	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 (class class class)co 7307 0cc0 10921 1c1 10922 + caddc 10924 · cmul 10926 2c2 12078 3c3 12079 4c4 12080 5c5 12081 6c6 12082 8c8 12084 cdc 12487 ↑cexp 13832

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-7 12091 df-8 12092 df-9 12093 df-n0 12284 df-z 12370 df-dec 12488 df-uz 12633 df-seq 13772 df-exp 13833

This theorem is referenced by: 1259lem1 16881 fmtno4 45248 ackval41 46285 ackval42 46286 ackval42a 46287

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