Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > blen1b | Structured version Visualization version GIF version |
Description: The binary length of a nonnegative integer is 1 if the integer is 0 or 1. (Contributed by AV, 30-May-2020.) |
Ref | Expression |
---|---|
blen1b | ⊢ (𝑁 ∈ ℕ0 → ((#b‘𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11902 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | blennn 44642 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) = ((⌊‘(2 logb 𝑁)) + 1)) | |
3 | 2 | eqeq1d 2825 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) = 1 ↔ ((⌊‘(2 logb 𝑁)) + 1) = 1)) |
4 | 2rp 12397 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ+ | |
5 | 4 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ+) |
6 | nnrp 12403 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
7 | 1ne2 11848 | . . . . . . . . . . . . 13 ⊢ 1 ≠ 2 | |
8 | 7 | necomi 3072 | . . . . . . . . . . . 12 ⊢ 2 ≠ 1 |
9 | 8 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1) |
10 | relogbcl 25353 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ∧ 2 ≠ 1) → (2 logb 𝑁) ∈ ℝ) | |
11 | 5, 6, 9, 10 | syl3anc 1367 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2 logb 𝑁) ∈ ℝ) |
12 | 11 | flcld 13171 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℤ) |
13 | 12 | zcnd 12091 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (⌊‘(2 logb 𝑁)) ∈ ℂ) |
14 | 1cnd 10638 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
15 | 13, 14, 14 | addlsub 11058 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (((⌊‘(2 logb 𝑁)) + 1) = 1 ↔ (⌊‘(2 logb 𝑁)) = (1 − 1))) |
16 | 1m1e0 11712 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (1 − 1) = 0) |
18 | 17 | eqeq2d 2834 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) = (1 − 1) ↔ (⌊‘(2 logb 𝑁)) = 0)) |
19 | 0z 11995 | . . . . . . . 8 ⊢ 0 ∈ ℤ | |
20 | flbi 13189 | . . . . . . . 8 ⊢ (((2 logb 𝑁) ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘(2 logb 𝑁)) = 0 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < (0 + 1)))) | |
21 | 11, 19, 20 | sylancl 588 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((⌊‘(2 logb 𝑁)) = 0 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < (0 + 1)))) |
22 | 15, 18, 21 | 3bitrd 307 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (((⌊‘(2 logb 𝑁)) + 1) = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < (0 + 1)))) |
23 | 0p1e1 11762 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
24 | 23 | breq2i 5076 | . . . . . . . 8 ⊢ ((2 logb 𝑁) < (0 + 1) ↔ (2 logb 𝑁) < 1) |
25 | 24 | anbi2i 624 | . . . . . . 7 ⊢ ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < (0 + 1)) ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) |
26 | nnlog2ge0lt1 44633 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) | |
27 | 26 | biimpar 480 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) → 𝑁 = 1) |
28 | 27 | olcd 870 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) → (𝑁 = 0 ∨ 𝑁 = 1)) |
29 | 28 | ex 415 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) → (𝑁 = 0 ∨ 𝑁 = 1))) |
30 | 25, 29 | syl5bi 244 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < (0 + 1)) → (𝑁 = 0 ∨ 𝑁 = 1))) |
31 | 22, 30 | sylbid 242 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (((⌊‘(2 logb 𝑁)) + 1) = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
32 | 3, 31 | sylbid 242 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
33 | orc 863 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 = 0 ∨ 𝑁 = 1)) | |
34 | 33 | a1d 25 | . . . 4 ⊢ (𝑁 = 0 → ((#b‘𝑁) = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
35 | 32, 34 | jaoi 853 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((#b‘𝑁) = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
36 | 1, 35 | sylbi 219 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((#b‘𝑁) = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
37 | fveq2 6672 | . . . 4 ⊢ (𝑁 = 0 → (#b‘𝑁) = (#b‘0)) | |
38 | blen0 44639 | . . . 4 ⊢ (#b‘0) = 1 | |
39 | 37, 38 | syl6eq 2874 | . . 3 ⊢ (𝑁 = 0 → (#b‘𝑁) = 1) |
40 | fveq2 6672 | . . . 4 ⊢ (𝑁 = 1 → (#b‘𝑁) = (#b‘1)) | |
41 | blen1 44651 | . . . 4 ⊢ (#b‘1) = 1 | |
42 | 40, 41 | syl6eq 2874 | . . 3 ⊢ (𝑁 = 1 → (#b‘𝑁) = 1) |
43 | 39, 42 | jaoi 853 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 = 1) → (#b‘𝑁) = 1) |
44 | 36, 43 | impbid1 227 | 1 ⊢ (𝑁 ∈ ℕ0 → ((#b‘𝑁) = 1 ↔ (𝑁 = 0 ∨ 𝑁 = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ℝ+crp 12392 ⌊cfl 13163 logb clogb 25344 #bcblen 44636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-logb 25345 df-blen 44637 |
This theorem is referenced by: nn0sumshdiglem2 44689 |
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