Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > blennn | Structured version Visualization version GIF version |
Description: The binary length of a positive integer. (Contributed by AV, 21-May-2020.) |
Ref | Expression |
---|---|
blennn | ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnne0 11752 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
2 | blenn0 45482 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 0) → (#b‘𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1)) | |
3 | 1, 2 | mpdan 687 | . 2 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb (abs‘𝑁))) + 1)) |
4 | nnre 11725 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
5 | nnnn0 11985 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
6 | 5 | nn0ge0d 12041 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
7 | 4, 6 | absidd 14874 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (abs‘𝑁) = 𝑁) |
8 | 7 | oveq2d 7188 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 logb (abs‘𝑁)) = (2 logb 𝑁)) |
9 | 8 | fveq2d 6680 | . . 3 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb (abs‘𝑁))) = (⌊‘(2 logb 𝑁))) |
10 | 9 | oveq1d 7187 | . 2 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb (abs‘𝑁))) + 1) = ((⌊‘(2 logb 𝑁)) + 1)) |
11 | 3, 10 | eqtrd 2773 | 1 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ‘cfv 6339 (class class class)co 7172 0cc0 10617 1c1 10618 + caddc 10620 ℕcn 11718 2c2 11773 ⌊cfl 13253 abscabs 14685 logb clogb 25504 #bcblen 45478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-sup 8981 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-n0 11979 df-z 12065 df-uz 12327 df-rp 12475 df-seq 13463 df-exp 13524 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-blen 45479 |
This theorem is referenced by: blennnelnn 45485 blenpw2 45487 blenpw2m1 45488 nnpw2blen 45489 blen1 45493 blen2 45494 blen1b 45497 blennnt2 45498 nnolog2flm1 45499 blennngt2o2 45501 blennn0e2 45503 dig2nn0ld 45513 dig2nn1st 45514 |
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