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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 11987 | . . . . 5 ⊢ (8 + 3) = ;11 | |
2 | 1 | eqcomi 2781 | . . . 4 ⊢ ;11 = (8 + 3) |
3 | 2 | oveq2i 6981 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
4 | 2cn 11508 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | 8nn0 11725 | . . . 4 ⊢ 8 ∈ ℕ0 | |
6 | 3nn0 11720 | . . . 4 ⊢ 3 ∈ ℕ0 | |
7 | expadd 13279 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
8 | 4, 5, 6, 7 | mp3an 1440 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
9 | 3, 8 | eqtri 2796 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
10 | 2exp8 16269 | . . . 4 ⊢ (2↑8) = ;;256 | |
11 | cu2 13371 | . . . 4 ⊢ (2↑3) = 8 | |
12 | 10, 11 | oveq12i 6982 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
13 | 2nn0 11719 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 5nn0 11722 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
15 | 13, 14 | deccl 11919 | . . . 4 ⊢ ;25 ∈ ℕ0 |
16 | 6nn0 11723 | . . . 4 ⊢ 6 ∈ ℕ0 | |
17 | eqid 2772 | . . . 4 ⊢ ;;256 = ;;256 | |
18 | 4nn0 11721 | . . . 4 ⊢ 4 ∈ ℕ0 | |
19 | 0nn0 11717 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
20 | 13, 19 | deccl 11919 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
21 | eqid 2772 | . . . . . 6 ⊢ ;25 = ;25 | |
22 | 1nn0 11718 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
23 | 8cn 11535 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
24 | 8t2e16 12021 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
25 | 23, 4, 24 | mulcomli 10441 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
26 | 1p1e2 11565 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 11978 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 11966 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
29 | 5cn 11523 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
30 | 8t5e40 12024 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
31 | 23, 29, 30 | mulcomli 10441 | . . . . . 6 ⊢ (5 · 8) = ;40 |
32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 11971 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
33 | 4cn 11519 | . . . . . 6 ⊢ 4 ∈ ℂ | |
34 | 33 | addid2i 10620 | . . . . 5 ⊢ (0 + 4) = 4 |
35 | 20, 19, 18, 32, 34 | decaddi 11965 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
36 | 6cn 11527 | . . . . 5 ⊢ 6 ∈ ℂ | |
37 | 8t6e48 12025 | . . . . 5 ⊢ (8 · 6) = ;48 | |
38 | 23, 36, 37 | mulcomli 10441 | . . . 4 ⊢ (6 · 8) = ;48 |
39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 11971 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
40 | 12, 39 | eqtri 2796 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
41 | 9, 40 | eqtri 2796 | 1 ⊢ (2↑;11) = ;;;2048 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 (class class class)co 6970 ℂcc 10325 0cc0 10327 1c1 10328 + caddc 10330 · cmul 10332 2c2 11488 3c3 11489 4c4 11490 5c5 11491 6c6 11492 8c8 11494 ℕ0cn0 11700 ;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: m11nprm 43074 |
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