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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 11915 | . 2 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 11917 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | 2 | nn0cni 11910 | . . 3 ⊢ 4 ∈ ℂ |
4 | 2cn 11713 | . . 3 ⊢ 2 ∈ ℂ | |
5 | 4t2e8 11806 | . . 3 ⊢ (4 · 2) = 8 | |
6 | 3, 4, 5 | mulcomli 10650 | . 2 ⊢ (2 · 4) = 8 |
7 | 2exp4 16421 | . 2 ⊢ (2↑4) = ;16 | |
8 | 1nn0 11914 | . . . 4 ⊢ 1 ∈ ℕ0 | |
9 | 6nn0 11919 | . . . 4 ⊢ 6 ∈ ℕ0 | |
10 | 8, 9 | deccl 12114 | . . 3 ⊢ ;16 ∈ ℕ0 |
11 | eqid 2821 | . . 3 ⊢ ;16 = ;16 | |
12 | 9nn0 11922 | . . 3 ⊢ 9 ∈ ℕ0 | |
13 | 10 | nn0cni 11910 | . . . . 5 ⊢ ;16 ∈ ℂ |
14 | 13 | mulid1i 10645 | . . . 4 ⊢ (;16 · 1) = ;16 |
15 | 1p1e2 11763 | . . . 4 ⊢ (1 + 1) = 2 | |
16 | 5nn0 11918 | . . . 4 ⊢ 5 ∈ ℕ0 | |
17 | 9cn 11738 | . . . . 5 ⊢ 9 ∈ ℂ | |
18 | 6cn 11729 | . . . . 5 ⊢ 6 ∈ ℂ | |
19 | 9p6e15 12190 | . . . . 5 ⊢ (9 + 6) = ;15 | |
20 | 17, 18, 19 | addcomli 10832 | . . . 4 ⊢ (6 + 9) = ;15 |
21 | 8, 9, 12, 14, 15, 16, 20 | decaddci 12160 | . . 3 ⊢ ((;16 · 1) + 9) = ;25 |
22 | 3nn0 11916 | . . . 4 ⊢ 3 ∈ ℕ0 | |
23 | 18 | mulid2i 10646 | . . . . . 6 ⊢ (1 · 6) = 6 |
24 | 23 | oveq1i 7166 | . . . . 5 ⊢ ((1 · 6) + 3) = (6 + 3) |
25 | 6p3e9 11798 | . . . . 5 ⊢ (6 + 3) = 9 | |
26 | 24, 25 | eqtri 2844 | . . . 4 ⊢ ((1 · 6) + 3) = 9 |
27 | 6t6e36 12207 | . . . 4 ⊢ (6 · 6) = ;36 | |
28 | 9, 8, 9, 11, 9, 22, 26, 27 | decmul1c 12164 | . . 3 ⊢ (;16 · 6) = ;96 |
29 | 10, 8, 9, 11, 9, 12, 21, 28 | decmul2c 12165 | . 2 ⊢ (;16 · ;16) = ;;256 |
30 | 1, 2, 6, 7, 29 | numexp2x 16415 | 1 ⊢ (2↑8) = ;;256 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7156 1c1 10538 + caddc 10540 · cmul 10542 2c2 11693 3c3 11694 4c4 11695 5c5 11696 6c6 11697 8c8 11699 9c9 11700 ;cdc 12099 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: 2exp16 16424 2503lem1 16470 quart1lem 25433 quart1 25434 fmtno3 43733 fmtno4sqrt 43753 2exp11 43785 |
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