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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 12167 | . . . . 5 ⊢ (8 + 3) = ;11 | |
2 | 1 | eqcomi 2807 | . . . 4 ⊢ ;11 = (8 + 3) |
3 | 2 | oveq2i 7146 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
4 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | 8nn0 11908 | . . . 4 ⊢ 8 ∈ ℕ0 | |
6 | 3nn0 11903 | . . . 4 ⊢ 3 ∈ ℕ0 | |
7 | expadd 13467 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
8 | 4, 5, 6, 7 | mp3an 1458 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
9 | 3, 8 | eqtri 2821 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
10 | 2exp8 16415 | . . . 4 ⊢ (2↑8) = ;;256 | |
11 | cu2 13559 | . . . 4 ⊢ (2↑3) = 8 | |
12 | 10, 11 | oveq12i 7147 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
13 | 2nn0 11902 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 5nn0 11905 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
15 | 13, 14 | deccl 12101 | . . . 4 ⊢ ;25 ∈ ℕ0 |
16 | 6nn0 11906 | . . . 4 ⊢ 6 ∈ ℕ0 | |
17 | eqid 2798 | . . . 4 ⊢ ;;256 = ;;256 | |
18 | 4nn0 11904 | . . . 4 ⊢ 4 ∈ ℕ0 | |
19 | 0nn0 11900 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
20 | 13, 19 | deccl 12101 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
21 | eqid 2798 | . . . . . 6 ⊢ ;25 = ;25 | |
22 | 1nn0 11901 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
23 | 8cn 11722 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
24 | 8t2e16 12201 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
25 | 23, 4, 24 | mulcomli 10639 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
26 | 1p1e2 11750 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12158 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 12147 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
29 | 5cn 11713 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
30 | 8t5e40 12204 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
31 | 23, 29, 30 | mulcomli 10639 | . . . . . 6 ⊢ (5 · 8) = ;40 |
32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 12151 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
33 | 4cn 11710 | . . . . . 6 ⊢ 4 ∈ ℂ | |
34 | 33 | addid2i 10817 | . . . . 5 ⊢ (0 + 4) = 4 |
35 | 20, 19, 18, 32, 34 | decaddi 12146 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
36 | 6cn 11716 | . . . . 5 ⊢ 6 ∈ ℂ | |
37 | 8t6e48 12205 | . . . . 5 ⊢ (8 · 6) = ;48 | |
38 | 23, 36, 37 | mulcomli 10639 | . . . 4 ⊢ (6 · 8) = ;48 |
39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 12151 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
40 | 12, 39 | eqtri 2821 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
41 | 9, 40 | eqtri 2821 | 1 ⊢ (2↑;11) = ;;;2048 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 8c8 11686 ℕ0cn0 11885 ;cdc 12086 ↑cexp 13425 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: m11nprm 44119 |
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