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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnlog2ge0lt1 | Structured version Visualization version GIF version |
Description: A positive integer is 1 iff its binary logarithm is between 0 and 1. (Contributed by AV, 30-May-2020.) |
Ref | Expression |
---|---|
nnlog2ge0lt1 | ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0le0 11737 | . . . . 5 ⊢ 0 ≤ 0 | |
2 | 2cn 11711 | . . . . . 6 ⊢ 2 ∈ ℂ | |
3 | 2ne0 11740 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 1ne2 11844 | . . . . . . 7 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 3070 | . . . . . 6 ⊢ 2 ≠ 1 |
6 | logb1 25346 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0) | |
7 | 2, 3, 5, 6 | mp3an 1457 | . . . . 5 ⊢ (2 logb 1) = 0 |
8 | 1, 7 | breqtrri 5092 | . . . 4 ⊢ 0 ≤ (2 logb 1) |
9 | 0lt1 11161 | . . . . 5 ⊢ 0 < 1 | |
10 | 7, 9 | eqbrtri 5086 | . . . 4 ⊢ (2 logb 1) < 1 |
11 | 8, 10 | pm3.2i 473 | . . 3 ⊢ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1) |
12 | oveq2 7163 | . . . . 5 ⊢ (𝑁 = 1 → (2 logb 𝑁) = (2 logb 1)) | |
13 | 12 | breq2d 5077 | . . . 4 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ↔ 0 ≤ (2 logb 1))) |
14 | 12 | breq1d 5075 | . . . 4 ⊢ (𝑁 = 1 → ((2 logb 𝑁) < 1 ↔ (2 logb 1) < 1)) |
15 | 13, 14 | anbi12d 632 | . . 3 ⊢ (𝑁 = 1 → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) ↔ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1))) |
16 | 11, 15 | mpbiri 260 | . 2 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) |
17 | 2z 12013 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
18 | uzid 12257 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
19 | 17, 18 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
20 | 19 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ (ℤ≥‘2)) |
21 | nnrp 12399 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
22 | logbge0b 44622 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) | |
23 | 20, 21, 22 | syl2anc 586 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) |
24 | logblt1b 44623 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) | |
25 | 20, 21, 24 | syl2anc 586 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) |
26 | 23, 25 | anbi12d 632 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) ↔ (1 ≤ 𝑁 ∧ 𝑁 < 2))) |
27 | df-2 11699 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
28 | 27 | breq2i 5073 | . . . . . . 7 ⊢ (𝑁 < 2 ↔ 𝑁 < (1 + 1)) |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 < 2 ↔ 𝑁 < (1 + 1))) |
30 | 29 | anbi2d 630 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
31 | nnre 11644 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
32 | 1zzd 12012 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
33 | flbi 13185 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) | |
34 | 31, 32, 33 | syl2anc 586 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
35 | 30, 34 | bitr4d 284 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (⌊‘𝑁) = 1)) |
36 | nnz 12003 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
37 | flid 13177 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
38 | 36, 37 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁) |
39 | 38 | eqcomd 2827 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 = (⌊‘𝑁)) |
40 | 39 | adantr 483 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = (⌊‘𝑁)) |
41 | simpr 487 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → (⌊‘𝑁) = 1) | |
42 | 40, 41 | eqtrd 2856 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = 1) |
43 | 42 | ex 415 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 → 𝑁 = 1)) |
44 | 35, 43 | sylbid 242 | . . 3 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) → 𝑁 = 1)) |
45 | 26, 44 | sylbid 242 | . 2 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) → 𝑁 = 1)) |
46 | 16, 45 | impbid2 228 | 1 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 ≤ cle 10675 ℕcn 11637 2c2 11691 ℤcz 11980 ℤ≥cuz 12242 ℝ+crp 12388 ⌊cfl 13159 logb clogb 25341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464 df-log 25139 df-logb 25342 |
This theorem is referenced by: blen1b 44647 |
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