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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 11719 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 11721 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2 | nn0cni 11713 | . . 3 ⊢ 4 ∈ ℂ |
4 | 2cn 11508 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4t2e8 11608 | . . 3 ⊢ (4 · 2) = 8 | |
6 | 3, 4, 5 | mulcomli 10441 | . 2 ⊢ (2 · 4) = 8 |
7 | 2exp4 16267 | . 2 ⊢ (2↑4) = ;16 | |
8 | 1nn0 11718 | . . . 4 ⊢ 1 ∈ ℕ0 | |
9 | 6nn0 11723 | . . . 4 ⊢ 6 ∈ ℕ0 | |
10 | 8, 9 | deccl 11919 | . . 3 ⊢ ;16 ∈ ℕ0 |
11 | eqid 2772 | . . 3 ⊢ ;16 = ;16 | |
12 | 9nn0 11726 | . . 3 ⊢ 9 ∈ ℕ0 | |
13 | 10 | nn0cni 11713 | . . . . 5 ⊢ ;16 ∈ ℂ |
14 | 13 | mulid1i 10436 | . . . 4 ⊢ (;16 · 1) = ;16 |
15 | 1p1e2 11565 | . . . 4 ⊢ (1 + 1) = 2 | |
16 | 5nn0 11722 | . . . 4 ⊢ 5 ∈ ℕ0 | |
17 | 9cn 11539 | . . . . 5 ⊢ 9 ∈ ℂ | |
18 | 6cn 11527 | . . . . 5 ⊢ 6 ∈ ℂ | |
19 | 9p6e15 11997 | . . . . 5 ⊢ (9 + 6) = ;15 | |
20 | 17, 18, 19 | addcomli 10624 | . . . 4 ⊢ (6 + 9) = ;15 |
21 | 8, 9, 12, 14, 15, 16, 20 | decaddci 11966 | . . 3 ⊢ ((;16 · 1) + 9) = ;25 |
22 | 3nn0 11720 | . . . 4 ⊢ 3 ∈ ℕ0 | |
23 | 18 | mulid2i 10437 | . . . . . 6 ⊢ (1 · 6) = 6 |
24 | 23 | oveq1i 6980 | . . . . 5 ⊢ ((1 · 6) + 3) = (6 + 3) |
25 | 6p3e9 11600 | . . . . 5 ⊢ (6 + 3) = 9 | |
26 | 24, 25 | eqtri 2796 | . . . 4 ⊢ ((1 · 6) + 3) = 9 |
27 | 6t6e36 12014 | . . . 4 ⊢ (6 · 6) = ;36 | |
28 | 9, 8, 9, 11, 9, 22, 26, 27 | decmul1c 11971 | . . 3 ⊢ (;16 · 6) = ;96 |
29 | 10, 8, 9, 11, 9, 12, 21, 28 | decmul2c 11972 | . 2 ⊢ (;16 · ;16) = ;;256 |
30 | 1, 2, 6, 7, 29 | numexp2x 16261 | 1 ⊢ (2↑8) = ;;256 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 (class class class)co 6970 1c1 10328 + caddc 10330 · cmul 10332 2c2 11488 3c3 11489 4c4 11490 5c5 11491 6c6 11492 8c8 11494 9c9 11495 ;cdc 11904 ↑cexp 13237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-seq 13178 df-exp 13238 |
This theorem is referenced by: 2exp16 16270 2503lem1 16316 quart1lem 25124 quart1 25125 fmtno3 43021 fmtno4sqrt 43041 2exp11 43073 |
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