2exp16 - Metamath Proof Explorer

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Theorem 2exp16 15635

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 11186	. 2 ⊢ 2 ∈ ℕ₀
2		8nn0 11192	. 2 ⊢ 8 ∈ ℕ₀
3		8cn 10983	. . 3 ⊢ 8 ∈ ℂ
4		2cn 10968	. . 3 ⊢ 2 ∈ ℂ
5		8t2e16 11530	. . 3 ⊢ (8 · 2) = ;16
6	3, 4, 5	mulcomli 9926	. 2 ⊢ (2 · 8) = ;16
7		2exp8 15634	. 2 ⊢ (2↑8) = ;;256
8		5nn0 11189	. . . . 5 ⊢ 5 ∈ ℕ₀
9	1, 8	deccl 11388	. . . 4 ⊢ ;25 ∈ ℕ₀
10		6nn0 11190	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 11388	. . 3 ⊢ ;;256 ∈ ℕ₀
12		eqid 2610	. . 3 ⊢ ;;256 = ;;256
13		1nn0 11185	. . . . 5 ⊢ 1 ∈ ℕ₀
14	13, 8	deccl 11388	. . . 4 ⊢ ;15 ∈ ℕ₀
15		3nn0 11187	. . . 4 ⊢ 3 ∈ ℕ₀
16	14, 15	deccl 11388	. . 3 ⊢ ;;153 ∈ ℕ₀
17		eqid 2610	. . . 4 ⊢ ;25 = ;25
18		eqid 2610	. . . 4 ⊢ ;;153 = ;;153
19	13, 1	deccl 11388	. . . . 5 ⊢ ;12 ∈ ℕ₀
20	19, 2	deccl 11388	. . . 4 ⊢ ;;128 ∈ ℕ₀
21		4nn0 11188	. . . . . 6 ⊢ 4 ∈ ℕ₀
22	13, 21	deccl 11388	. . . . 5 ⊢ ;14 ∈ ℕ₀
23		eqid 2610	. . . . . 6 ⊢ ;15 = ;15
24		eqid 2610	. . . . . 6 ⊢ ;;128 = ;;128
25		0nn0 11184	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
26	13	dec0h 11398	. . . . . . . 8 ⊢ 1 = ;01
27		eqid 2610	. . . . . . . 8 ⊢ ;12 = ;12
28		0p1e1 11009	. . . . . . . 8 ⊢ (0 + 1) = 1
29		1p2e3 11029	. . . . . . . 8 ⊢ (1 + 2) = 3
30	25, 13, 13, 1, 26, 27, 28, 29	decadd 11446	. . . . . . 7 ⊢ (1 + ;12) = ;13
31		3p1e4 11030	. . . . . . 7 ⊢ (3 + 1) = 4
32	13, 15, 13, 30, 31	decaddi 11455	. . . . . 6 ⊢ ((1 + ;12) + 1) = ;14
33		5cn 10977	. . . . . . 7 ⊢ 5 ∈ ℂ
34		8p5e13 11491	. . . . . . 7 ⊢ (8 + 5) = ;13
35	3, 33, 34	addcomli 10107	. . . . . 6 ⊢ (5 + 8) = ;13
36	13, 8, 19, 2, 23, 24, 32, 15, 35	decaddc 11448	. . . . 5 ⊢ (;15 + ;;128) = ;;143
37		eqid 2610	. . . . . . 7 ⊢ ;14 = ;14
38		4p1e5 11031	. . . . . . 7 ⊢ (4 + 1) = 5
39	13, 21, 13, 37, 38	decaddi 11455	. . . . . 6 ⊢ (;14 + 1) = ;15
40		2t2e4 11054	. . . . . . . 8 ⊢ (2 · 2) = 4
41		1p1e2 11011	. . . . . . . 8 ⊢ (1 + 1) = 2
42	40, 41	oveq12i 6561	. . . . . . 7 ⊢ ((2 · 2) + (1 + 1)) = (4 + 2)
43		4p2e6 11039	. . . . . . 7 ⊢ (4 + 2) = 6
44	42, 43	eqtri 2632	. . . . . 6 ⊢ ((2 · 2) + (1 + 1)) = 6
45		5t2e10 11510	. . . . . . 7 ⊢ (5 · 2) = ;10
46	33	addid2i 10103	. . . . . . 7 ⊢ (0 + 5) = 5
47	13, 25, 8, 45, 46	decaddi 11455	. . . . . 6 ⊢ ((5 · 2) + 5) = ;15
48	1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47	decmac 11442	. . . . 5 ⊢ ((;25 · 2) + (;14 + 1)) = ;65
49		6t2e12 11517	. . . . . 6 ⊢ (6 · 2) = ;12
50		3cn 10972	. . . . . . 7 ⊢ 3 ∈ ℂ
51		3p2e5 11037	. . . . . . 7 ⊢ (3 + 2) = 5
52	50, 4, 51	addcomli 10107	. . . . . 6 ⊢ (2 + 3) = 5
53	13, 1, 15, 49, 52	decaddi 11455	. . . . 5 ⊢ ((6 · 2) + 3) = ;15
54	9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53	decmac 11442	. . . 4 ⊢ ((;;256 · 2) + (;15 + ;;128)) = ;;655
55	15	dec0h 11398	. . . . 5 ⊢ 3 = ;03
56	50	addid2i 10103	. . . . . . 7 ⊢ (0 + 3) = 3
57	56, 55	eqtri 2632	. . . . . 6 ⊢ (0 + 3) = ;03
58	4	addid2i 10103	. . . . . . . 8 ⊢ (0 + 2) = 2
59	58	oveq2i 6560	. . . . . . 7 ⊢ ((2 · 5) + (0 + 2)) = ((2 · 5) + 2)
60	33, 4, 45	mulcomli 9926	. . . . . . . 8 ⊢ (2 · 5) = ;10
61	13, 25, 1, 60, 58	decaddi 11455	. . . . . . 7 ⊢ ((2 · 5) + 2) = ;12
62	59, 61	eqtri 2632	. . . . . 6 ⊢ ((2 · 5) + (0 + 2)) = ;12
63		5t5e25 11515	. . . . . . 7 ⊢ (5 · 5) = ;25
64		5p3e8 11043	. . . . . . 7 ⊢ (5 + 3) = 8
65	1, 8, 15, 63, 64	decaddi 11455	. . . . . 6 ⊢ ((5 · 5) + 3) = ;28
66	1, 8, 25, 15, 17, 57, 8, 2, 1, 62, 65	decmac 11442	. . . . 5 ⊢ ((;25 · 5) + (0 + 3)) = ;;128
67		6t5e30 11520	. . . . . 6 ⊢ (6 · 5) = ;30
68	15, 25, 15, 67, 56	decaddi 11455	. . . . 5 ⊢ ((6 · 5) + 3) = ;33
69	9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68	decmac 11442	. . . 4 ⊢ ((;;256 · 5) + 3) = ;;;1283
70	1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69	decma2c 11444	. . 3 ⊢ ((;;256 · ;25) + ;;153) = ;;;6553
71		6cn 10979	. . . . . . 7 ⊢ 6 ∈ ℂ
72	71, 4, 49	mulcomli 9926	. . . . . 6 ⊢ (2 · 6) = ;12
73	13, 1, 15, 72, 52	decaddi 11455	. . . . 5 ⊢ ((2 · 6) + 3) = ;15
74	71, 33, 67	mulcomli 9926	. . . . . 6 ⊢ (5 · 6) = ;30
75	15, 25, 15, 74, 56	decaddi 11455	. . . . 5 ⊢ ((5 · 6) + 3) = ;33
76	1, 8, 15, 17, 10, 15, 15, 73, 75	decrmac 11453	. . . 4 ⊢ ((;25 · 6) + 3) = ;;153
77		6t6e36 11522	. . . 4 ⊢ (6 · 6) = ;36
78	10, 9, 10, 12, 10, 15, 76, 77	decmul1c 11463	. . 3 ⊢ (;;256 · 6) = ;;;1536
79	11, 9, 10, 12, 10, 16, 70, 78	decmul2c 11465	. 2 ⊢ (;;256 · ;;256) = ;;;;65536
80	1, 2, 6, 7, 79	numexp2x 15621	1 ⊢ (2↑;16) = ;;;;65536

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 8c8 10953 cdc 11369 ↑cexp 12722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-seq 12664 df-exp 12723

This theorem is referenced by: 1259lem1 15676 fmtno4 40002

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