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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 12426 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 12428 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2 | nn0cni 12421 | . . 3 ⊢ 4 ∈ ℂ |
4 | 2cn 12224 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4t2e8 12317 | . . 3 ⊢ (4 · 2) = 8 | |
6 | 3, 4, 5 | mulcomli 11160 | . 2 ⊢ (2 · 4) = 8 |
7 | 2exp4 16949 | . 2 ⊢ (2↑4) = ;16 | |
8 | 1nn0 12425 | . . . 4 ⊢ 1 ∈ ℕ0 | |
9 | 6nn0 12430 | . . . 4 ⊢ 6 ∈ ℕ0 | |
10 | 8, 9 | deccl 12629 | . . 3 ⊢ ;16 ∈ ℕ0 |
11 | eqid 2736 | . . 3 ⊢ ;16 = ;16 | |
12 | 9nn0 12433 | . . 3 ⊢ 9 ∈ ℕ0 | |
13 | 10 | nn0cni 12421 | . . . . 5 ⊢ ;16 ∈ ℂ |
14 | 13 | mulid1i 11155 | . . . 4 ⊢ (;16 · 1) = ;16 |
15 | 1p1e2 12274 | . . . 4 ⊢ (1 + 1) = 2 | |
16 | 5nn0 12429 | . . . 4 ⊢ 5 ∈ ℕ0 | |
17 | 9cn 12249 | . . . . 5 ⊢ 9 ∈ ℂ | |
18 | 6cn 12240 | . . . . 5 ⊢ 6 ∈ ℂ | |
19 | 9p6e15 12705 | . . . . 5 ⊢ (9 + 6) = ;15 | |
20 | 17, 18, 19 | addcomli 11343 | . . . 4 ⊢ (6 + 9) = ;15 |
21 | 8, 9, 12, 14, 15, 16, 20 | decaddci 12675 | . . 3 ⊢ ((;16 · 1) + 9) = ;25 |
22 | 3nn0 12427 | . . . 4 ⊢ 3 ∈ ℕ0 | |
23 | 18 | mulid2i 11156 | . . . . . 6 ⊢ (1 · 6) = 6 |
24 | 23 | oveq1i 7363 | . . . . 5 ⊢ ((1 · 6) + 3) = (6 + 3) |
25 | 6p3e9 12309 | . . . . 5 ⊢ (6 + 3) = 9 | |
26 | 24, 25 | eqtri 2764 | . . . 4 ⊢ ((1 · 6) + 3) = 9 |
27 | 6t6e36 12722 | . . . 4 ⊢ (6 · 6) = ;36 | |
28 | 9, 8, 9, 11, 9, 22, 26, 27 | decmul1c 12679 | . . 3 ⊢ (;16 · 6) = ;96 |
29 | 10, 8, 9, 11, 9, 12, 21, 28 | decmul2c 12680 | . 2 ⊢ (;16 · ;16) = ;;256 |
30 | 1, 2, 6, 7, 29 | numexp2x 16943 | 1 ⊢ (2↑8) = ;;256 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7353 1c1 11048 + caddc 11050 · cmul 11052 2c2 12204 3c3 12205 4c4 12206 5c5 12207 6c6 12208 8c8 12210 9c9 12211 ;cdc 12614 ↑cexp 13959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-seq 13899 df-exp 13960 |
This theorem is referenced by: 2exp11 16954 2exp16 16955 2503lem1 17001 quart1lem 26189 quart1 26190 lcmineqlem 40476 aks4d1p1 40500 fmtno3 45675 fmtno4sqrt 45695 |
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