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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 11421 | . . . . 5 ⊢ 0 ≤ 0 | |
2 | 2cn 11388 | . . . . . 6 ⊢ 2 ∈ ℂ | |
3 | 2ne0 11424 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 1ne2 11528 | . . . . . . 7 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 3025 | . . . . . 6 ⊢ 2 ≠ 1 |
6 | logb1 24851 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0) | |
7 | 2, 3, 5, 6 | mp3an 1586 | . . . . 5 ⊢ (2 logb 1) = 0 |
8 | 1, 7 | breqtrri 4870 | . . . 4 ⊢ 0 ≤ (2 logb 1) |
9 | 0lt1 10842 | . . . . 5 ⊢ 0 < 1 | |
10 | 7, 9 | eqbrtri 4864 | . . . 4 ⊢ (2 logb 1) < 1 |
11 | 8, 10 | pm3.2i 463 | . . 3 ⊢ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1) |
12 | oveq2 6886 | . . . . 5 ⊢ (𝑁 = 1 → (2 logb 𝑁) = (2 logb 1)) | |
13 | 12 | breq2d 4855 | . . . 4 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ↔ 0 ≤ (2 logb 1))) |
14 | 12 | breq1d 4853 | . . . 4 ⊢ (𝑁 = 1 → ((2 logb 𝑁) < 1 ↔ (2 logb 1) < 1)) |
15 | 13, 14 | anbi12d 625 | . . 3 ⊢ (𝑁 = 1 → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) ↔ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1))) |
16 | 11, 15 | mpbiri 250 | . 2 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1)) |
17 | 2z 11699 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
18 | uzid 11945 | . . . . . . 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 12087 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
22 | logbge0b 43156 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) | |
23 | 20, 21, 22 | syl2anc 580 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) |
24 | logblt1b 43157 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) | |
25 | 20, 21, 24 | syl2anc 580 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) |
26 | 23, 25 | anbi12d 625 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) ↔ (1 ≤ 𝑁 ∧ 𝑁 < 2))) |
27 | df-2 11376 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
28 | 27 | breq2i 4851 | . . . . . . 7 ⊢ (𝑁 < 2 ↔ 𝑁 < (1 + 1)) |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 < 2 ↔ 𝑁 < (1 + 1))) |
30 | 29 | anbi2d 623 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
31 | nnre 11320 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
32 | 1zzd 11698 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
33 | flbi 12872 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) | |
34 | 31, 32, 33 | syl2anc 580 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
35 | 30, 34 | bitr4d 274 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (⌊‘𝑁) = 1)) |
36 | nnz 11689 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
37 | flid 12864 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
38 | 36, 37 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁) |
39 | 38 | eqcomd 2805 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 = (⌊‘𝑁)) |
40 | 39 | adantr 473 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = (⌊‘𝑁)) |
41 | simpr 478 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → (⌊‘𝑁) = 1) | |
42 | 40, 41 | eqtrd 2833 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = 1) |
43 | 42 | ex 402 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((⌊‘𝑁) = 1 → 𝑁 = 1)) |
44 | 35, 43 | sylbid 232 | . . 3 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) → 𝑁 = 1)) |
45 | 26, 44 | sylbid 232 | . 2 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) → 𝑁 = 1)) |
46 | 16, 45 | impbid2 218 | 1 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ↔ (0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 < clt 10363 ≤ cle 10364 ℕcn 11312 2c2 11368 ℤcz 11666 ℤ≥cuz 11930 ℝ+crp 12074 ⌊cfl 12846 logb clogb 24846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ioc 12429 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-mod 12924 df-seq 13056 df-exp 13115 df-fac 13314 df-bc 13343 df-hash 13371 df-shft 14148 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-sum 14758 df-ef 15134 df-sin 15136 df-cos 15137 df-pi 15139 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-fbas 20065 df-fg 20066 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-lp 21269 df-perf 21270 df-cn 21360 df-cnp 21361 df-haus 21448 df-tx 21694 df-hmeo 21887 df-fil 21978 df-fm 22070 df-flim 22071 df-flf 22072 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-limc 23971 df-dv 23972 df-log 24644 df-logb 24847 |
This theorem is referenced by: blen1b 43181 |
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