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Mirrors > Home > MPE Home > Th. List > Mathboxes > blengt1fldiv2p1 | Structured version Visualization version GIF version |
Description: The binary length of an integer greater than 1 is the binary length of the integer divided by 2, increased by one. (Contributed by AV, 3-Jun-2020.) |
Ref | Expression |
---|---|
blengt1fldiv2p1 | ⊢ (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2nn 12463 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
2 | nneop 45499 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) |
4 | nnnn0 12080 | . . . . . . . . 9 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℕ0) | |
5 | blennn0em1 45564 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ0) → (#b‘(𝑁 / 2)) = ((#b‘𝑁) − 1)) | |
6 | 4, 5 | sylan2 596 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → (#b‘(𝑁 / 2)) = ((#b‘𝑁) − 1)) |
7 | 6 | ancoms 462 | . . . . . . 7 ⊢ (((𝑁 / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (#b‘(𝑁 / 2)) = ((#b‘𝑁) − 1)) |
8 | 7 | oveq1d 7217 | . . . . . 6 ⊢ (((𝑁 / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((#b‘(𝑁 / 2)) + 1) = (((#b‘𝑁) − 1) + 1)) |
9 | nnz 12182 | . . . . . . . . . . 11 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ) | |
10 | flid 13366 | . . . . . . . . . . 11 ⊢ ((𝑁 / 2) ∈ ℤ → (⌊‘(𝑁 / 2)) = (𝑁 / 2)) | |
11 | 9, 10 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑁 / 2) ∈ ℕ → (⌊‘(𝑁 / 2)) = (𝑁 / 2)) |
12 | 11 | eqcomd 2740 | . . . . . . . . 9 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) = (⌊‘(𝑁 / 2))) |
13 | 12 | fveq2d 6710 | . . . . . . . 8 ⊢ ((𝑁 / 2) ∈ ℕ → (#b‘(𝑁 / 2)) = (#b‘(⌊‘(𝑁 / 2)))) |
14 | 13 | oveq1d 7217 | . . . . . . 7 ⊢ ((𝑁 / 2) ∈ ℕ → ((#b‘(𝑁 / 2)) + 1) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
15 | 14 | adantr 484 | . . . . . 6 ⊢ (((𝑁 / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((#b‘(𝑁 / 2)) + 1) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
16 | blennnelnn 45549 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | |
17 | 16 | nncnd 11829 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℂ) |
18 | npcan1 11240 | . . . . . . . 8 ⊢ ((#b‘𝑁) ∈ ℂ → (((#b‘𝑁) − 1) + 1) = (#b‘𝑁)) | |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((#b‘𝑁) − 1) + 1) = (#b‘𝑁)) |
20 | 19 | adantl 485 | . . . . . 6 ⊢ (((𝑁 / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((#b‘𝑁) − 1) + 1) = (#b‘𝑁)) |
21 | 8, 15, 20 | 3eqtr3rd 2783 | . . . . 5 ⊢ (((𝑁 / 2) ∈ ℕ ∧ 𝑁 ∈ ℕ) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
22 | 21 | expcom 417 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))) |
23 | 22, 1 | syl11 33 | . . 3 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))) |
24 | nnnn0 12080 | . . . . . . 7 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℕ0) | |
25 | blennngt2o2 45565 | . . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (#b‘𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1)) | |
26 | 24, 25 | sylan2 596 | . . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ) → (#b‘𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1)) |
27 | 26 | ancoms 462 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → (#b‘𝑁) = ((#b‘((𝑁 − 1) / 2)) + 1)) |
28 | eluzge2nn0 12466 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ0) | |
29 | nn0ofldiv2 45505 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) | |
30 | 28, 24, 29 | syl2anr 600 | . . . . . . . 8 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2)) |
31 | 30 | eqcomd 2740 | . . . . . . 7 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → ((𝑁 − 1) / 2) = (⌊‘(𝑁 / 2))) |
32 | 31 | fveq2d 6710 | . . . . . 6 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → (#b‘((𝑁 − 1) / 2)) = (#b‘(⌊‘(𝑁 / 2)))) |
33 | 32 | oveq1d 7217 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → ((#b‘((𝑁 − 1) / 2)) + 1) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
34 | 27, 33 | eqtrd 2774 | . . . 4 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘2)) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
35 | 34 | ex 416 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))) |
36 | 23, 35 | jaoi 857 | . 2 ⊢ (((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ) → (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1))) |
37 | 3, 36 | mpcom 38 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → (#b‘𝑁) = ((#b‘(⌊‘(𝑁 / 2))) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 1c1 10713 + caddc 10715 − cmin 11045 / cdiv 11472 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ℤcz 12159 ℤ≥cuz 12421 ⌊cfl 13348 #bcblen 45542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-ef 15610 df-sin 15612 df-cos 15613 df-pi 15615 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-log 25417 df-cxp 25418 df-logb 25620 df-blen 45543 |
This theorem is referenced by: (None) |
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