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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 12261 | . . . . 5 ⊢ 0 ≤ 0 | |
2 | 2cn 12235 | . . . . . 6 ⊢ 2 ∈ ℂ | |
3 | 2ne0 12264 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 1ne2 12368 | . . . . . . 7 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 2999 | . . . . . 6 ⊢ 2 ≠ 1 |
6 | logb1 26135 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0) | |
7 | 2, 3, 5, 6 | mp3an 1462 | . . . . 5 ⊢ (2 logb 1) = 0 |
8 | 1, 7 | breqtrri 5137 | . . . 4 ⊢ 0 ≤ (2 logb 1) |
9 | 0lt1 11684 | . . . . 5 ⊢ 0 < 1 | |
10 | 7, 9 | eqbrtri 5131 | . . . 4 ⊢ (2 logb 1) < 1 |
11 | 8, 10 | pm3.2i 472 | . . 3 ⊢ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1) |
12 | oveq2 7370 | . . . . 5 ⊢ (𝑁 = 1 → (2 logb 𝑁) = (2 logb 1)) | |
13 | 12 | breq2d 5122 | . . . 4 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ↔ 0 ≤ (2 logb 1))) |
14 | 12 | breq1d 5120 | . . . 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 258 | . 2 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) |
17 | 2z 12542 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
18 | uzid 12785 | . . . . . . 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 12933 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
22 | logbge0b 46723 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) | |
23 | 20, 21, 22 | syl2anc 585 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) |
24 | logblt1b 46724 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) | |
25 | 20, 21, 24 | syl2anc 585 | . . . 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 12223 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
28 | 27 | breq2i 5118 | . . . . . . 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 12167 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
32 | 1zzd 12541 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
33 | flbi 13728 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) | |
34 | 31, 32, 33 | syl2anc 585 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
35 | 30, 34 | bitr4d 282 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (⌊‘𝑁) = 1)) |
36 | nnz 12527 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
37 | flid 13720 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
38 | 36, 37 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁) |
39 | 38 | eqcomd 2743 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 = (⌊‘𝑁)) |
40 | 39 | adantr 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = (⌊‘𝑁)) |
41 | simpr 486 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → (⌊‘𝑁) = 1) | |
42 | 40, 41 | eqtrd 2777 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = 1) |
43 | 42 | ex 414 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 → 𝑁 = 1)) |
44 | 35, 43 | sylbid 239 | . . 3 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) → 𝑁 = 1)) |
45 | 26, 44 | sylbid 239 | . 2 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) → 𝑁 = 1)) |
46 | 16, 45 | impbid2 225 | 1 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 ℕcn 12160 2c2 12215 ℤcz 12506 ℤ≥cuz 12770 ℝ+crp 12922 ⌊cfl 13702 logb clogb 26130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 df-log 25928 df-logb 26131 |
This theorem is referenced by: blen1b 46748 |
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