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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 12541 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 12543 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2 | nn0cni 12536 | . . 3 ⊢ 4 ∈ ℂ |
4 | 2cn 12339 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4t2e8 12432 | . . 3 ⊢ (4 · 2) = 8 | |
6 | 3, 4, 5 | mulcomli 11273 | . 2 ⊢ (2 · 4) = 8 |
7 | 2exp4 17087 | . 2 ⊢ (2↑4) = ;16 | |
8 | 1nn0 12540 | . . . 4 ⊢ 1 ∈ ℕ0 | |
9 | 6nn0 12545 | . . . 4 ⊢ 6 ∈ ℕ0 | |
10 | 8, 9 | deccl 12744 | . . 3 ⊢ ;16 ∈ ℕ0 |
11 | eqid 2726 | . . 3 ⊢ ;16 = ;16 | |
12 | 9nn0 12548 | . . 3 ⊢ 9 ∈ ℕ0 | |
13 | 10 | nn0cni 12536 | . . . . 5 ⊢ ;16 ∈ ℂ |
14 | 13 | mulridi 11268 | . . . 4 ⊢ (;16 · 1) = ;16 |
15 | 1p1e2 12389 | . . . 4 ⊢ (1 + 1) = 2 | |
16 | 5nn0 12544 | . . . 4 ⊢ 5 ∈ ℕ0 | |
17 | 9cn 12364 | . . . . 5 ⊢ 9 ∈ ℂ | |
18 | 6cn 12355 | . . . . 5 ⊢ 6 ∈ ℂ | |
19 | 9p6e15 12820 | . . . . 5 ⊢ (9 + 6) = ;15 | |
20 | 17, 18, 19 | addcomli 11456 | . . . 4 ⊢ (6 + 9) = ;15 |
21 | 8, 9, 12, 14, 15, 16, 20 | decaddci 12790 | . . 3 ⊢ ((;16 · 1) + 9) = ;25 |
22 | 3nn0 12542 | . . . 4 ⊢ 3 ∈ ℕ0 | |
23 | 18 | mullidi 11269 | . . . . . 6 ⊢ (1 · 6) = 6 |
24 | 23 | oveq1i 7434 | . . . . 5 ⊢ ((1 · 6) + 3) = (6 + 3) |
25 | 6p3e9 12424 | . . . . 5 ⊢ (6 + 3) = 9 | |
26 | 24, 25 | eqtri 2754 | . . . 4 ⊢ ((1 · 6) + 3) = 9 |
27 | 6t6e36 12837 | . . . 4 ⊢ (6 · 6) = ;36 | |
28 | 9, 8, 9, 11, 9, 22, 26, 27 | decmul1c 12794 | . . 3 ⊢ (;16 · 6) = ;96 |
29 | 10, 8, 9, 11, 9, 12, 21, 28 | decmul2c 12795 | . 2 ⊢ (;16 · ;16) = ;;256 |
30 | 1, 2, 6, 7, 29 | numexp2x 17081 | 1 ⊢ (2↑8) = ;;256 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7424 1c1 11159 + caddc 11161 · cmul 11163 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 8c8 12325 9c9 12326 ;cdc 12729 ↑cexp 14081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-seq 14022 df-exp 14082 |
This theorem is referenced by: 2exp11 17092 2exp16 17093 2503lem1 17139 quart1lem 26883 quart1 26884 lcmineqlem 41751 aks4d1p1 41775 fmtno3 47123 fmtno4sqrt 47143 |
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