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Mirrors > Home > MPE Home > Th. List > 2exp8 | Structured version Visualization version GIF version |
Description: Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.) |
Ref | Expression |
---|---|
2exp8 | ⊢ (2↑8) = ;;256 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12104 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 12106 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2 | nn0cni 12099 | . . 3 ⊢ 4 ∈ ℂ |
4 | 2cn 11902 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4t2e8 11995 | . . 3 ⊢ (4 · 2) = 8 | |
6 | 3, 4, 5 | mulcomli 10839 | . 2 ⊢ (2 · 4) = 8 |
7 | 2exp4 16635 | . 2 ⊢ (2↑4) = ;16 | |
8 | 1nn0 12103 | . . . 4 ⊢ 1 ∈ ℕ0 | |
9 | 6nn0 12108 | . . . 4 ⊢ 6 ∈ ℕ0 | |
10 | 8, 9 | deccl 12305 | . . 3 ⊢ ;16 ∈ ℕ0 |
11 | eqid 2737 | . . 3 ⊢ ;16 = ;16 | |
12 | 9nn0 12111 | . . 3 ⊢ 9 ∈ ℕ0 | |
13 | 10 | nn0cni 12099 | . . . . 5 ⊢ ;16 ∈ ℂ |
14 | 13 | mulid1i 10834 | . . . 4 ⊢ (;16 · 1) = ;16 |
15 | 1p1e2 11952 | . . . 4 ⊢ (1 + 1) = 2 | |
16 | 5nn0 12107 | . . . 4 ⊢ 5 ∈ ℕ0 | |
17 | 9cn 11927 | . . . . 5 ⊢ 9 ∈ ℂ | |
18 | 6cn 11918 | . . . . 5 ⊢ 6 ∈ ℂ | |
19 | 9p6e15 12381 | . . . . 5 ⊢ (9 + 6) = ;15 | |
20 | 17, 18, 19 | addcomli 11021 | . . . 4 ⊢ (6 + 9) = ;15 |
21 | 8, 9, 12, 14, 15, 16, 20 | decaddci 12351 | . . 3 ⊢ ((;16 · 1) + 9) = ;25 |
22 | 3nn0 12105 | . . . 4 ⊢ 3 ∈ ℕ0 | |
23 | 18 | mulid2i 10835 | . . . . . 6 ⊢ (1 · 6) = 6 |
24 | 23 | oveq1i 7220 | . . . . 5 ⊢ ((1 · 6) + 3) = (6 + 3) |
25 | 6p3e9 11987 | . . . . 5 ⊢ (6 + 3) = 9 | |
26 | 24, 25 | eqtri 2765 | . . . 4 ⊢ ((1 · 6) + 3) = 9 |
27 | 6t6e36 12398 | . . . 4 ⊢ (6 · 6) = ;36 | |
28 | 9, 8, 9, 11, 9, 22, 26, 27 | decmul1c 12355 | . . 3 ⊢ (;16 · 6) = ;96 |
29 | 10, 8, 9, 11, 9, 12, 21, 28 | decmul2c 12356 | . 2 ⊢ (;16 · ;16) = ;;256 |
30 | 1, 2, 6, 7, 29 | numexp2x 16629 | 1 ⊢ (2↑8) = ;;256 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 (class class class)co 7210 1c1 10727 + caddc 10729 · cmul 10731 2c2 11882 3c3 11883 4c4 11884 5c5 11885 6c6 11886 8c8 11888 9c9 11889 ;cdc 12290 ↑cexp 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-seq 13572 df-exp 13633 |
This theorem is referenced by: 2exp11 16640 2exp16 16641 2503lem1 16687 quart1lem 25735 quart1 25736 lcmineqlem 39792 aks4d1p1 39815 fmtno3 44674 fmtno4sqrt 44694 |
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