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Mirrors > Home > MPE Home > Th. List > 2exp11 | Structured version Visualization version GIF version |
Description: Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.) |
Ref | Expression |
---|---|
2exp11 | ⊢ (2↑;11) = ;;;2048 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8p3e11 12791 | . . . . 5 ⊢ (8 + 3) = ;11 | |
2 | 1 | eqcomi 2734 | . . . 4 ⊢ ;11 = (8 + 3) |
3 | 2 | oveq2i 7430 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
4 | 2cn 12320 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | 8nn0 12528 | . . . 4 ⊢ 8 ∈ ℕ0 | |
6 | 3nn0 12523 | . . . 4 ⊢ 3 ∈ ℕ0 | |
7 | expadd 14105 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
8 | 4, 5, 6, 7 | mp3an 1457 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
9 | 3, 8 | eqtri 2753 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
10 | 2exp8 17061 | . . . 4 ⊢ (2↑8) = ;;256 | |
11 | cu2 14199 | . . . 4 ⊢ (2↑3) = 8 | |
12 | 10, 11 | oveq12i 7431 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
13 | 2nn0 12522 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 5nn0 12525 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
15 | 13, 14 | deccl 12725 | . . . 4 ⊢ ;25 ∈ ℕ0 |
16 | 6nn0 12526 | . . . 4 ⊢ 6 ∈ ℕ0 | |
17 | eqid 2725 | . . . 4 ⊢ ;;256 = ;;256 | |
18 | 4nn0 12524 | . . . 4 ⊢ 4 ∈ ℕ0 | |
19 | 0nn0 12520 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
20 | 13, 19 | deccl 12725 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
21 | eqid 2725 | . . . . . 6 ⊢ ;25 = ;25 | |
22 | 1nn0 12521 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
23 | 8cn 12342 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
24 | 8t2e16 12825 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
25 | 23, 4, 24 | mulcomli 11255 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
26 | 1p1e2 12370 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12782 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 12771 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
29 | 5cn 12333 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
30 | 8t5e40 12828 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
31 | 23, 29, 30 | mulcomli 11255 | . . . . . 6 ⊢ (5 · 8) = ;40 |
32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 12775 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
33 | 4cn 12330 | . . . . . 6 ⊢ 4 ∈ ℂ | |
34 | 33 | addlidi 11434 | . . . . 5 ⊢ (0 + 4) = 4 |
35 | 20, 19, 18, 32, 34 | decaddi 12770 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
36 | 6cn 12336 | . . . . 5 ⊢ 6 ∈ ℂ | |
37 | 8t6e48 12829 | . . . . 5 ⊢ (8 · 6) = ;48 | |
38 | 23, 36, 37 | mulcomli 11255 | . . . 4 ⊢ (6 · 8) = ;48 |
39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 12775 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
40 | 12, 39 | eqtri 2753 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
41 | 9, 40 | eqtri 2753 | 1 ⊢ (2↑;11) = ;;;2048 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 2c2 12300 3c3 12301 4c4 12302 5c5 12303 6c6 12304 8c8 12306 ℕ0cn0 12505 ;cdc 12710 ↑cexp 14062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-seq 14003 df-exp 14063 |
This theorem is referenced by: 3lexlogpow5ineq2 41655 m11nprm 47075 |
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