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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 12329 | . . . . 5 ⊢ 0 ≤ 0 | |
2 | 2cn 12303 | . . . . . 6 ⊢ 2 ∈ ℂ | |
3 | 2ne0 12332 | . . . . . 6 ⊢ 2 ≠ 0 | |
4 | 1ne2 12436 | . . . . . . 7 ⊢ 1 ≠ 2 | |
5 | 4 | necomi 2990 | . . . . . 6 ⊢ 2 ≠ 1 |
6 | logb1 26675 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 logb 1) = 0) | |
7 | 2, 3, 5, 6 | mp3an 1458 | . . . . 5 ⊢ (2 logb 1) = 0 |
8 | 1, 7 | breqtrri 5169 | . . . 4 ⊢ 0 ≤ (2 logb 1) |
9 | 0lt1 11752 | . . . . 5 ⊢ 0 < 1 | |
10 | 7, 9 | eqbrtri 5163 | . . . 4 ⊢ (2 logb 1) < 1 |
11 | 8, 10 | pm3.2i 470 | . . 3 ⊢ (0 ≤ (2 logb 1) ∧ (2 logb 1) < 1) |
12 | oveq2 7422 | . . . . 5 ⊢ (𝑁 = 1 → (2 logb 𝑁) = (2 logb 1)) | |
13 | 12 | breq2d 5154 | . . . 4 ⊢ (𝑁 = 1 → (0 ≤ (2 logb 𝑁) ↔ 0 ≤ (2 logb 1))) |
14 | 12 | breq1d 5152 | . . . 4 ⊢ (𝑁 = 1 → ((2 logb 𝑁) < 1 ↔ (2 logb 1) < 1)) |
15 | 13, 14 | anbi12d 630 | . . 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 12610 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
18 | uzid 12853 | . . . . . . 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 13003 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
22 | logbge0b 47549 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) | |
23 | 20, 21, 22 | syl2anc 583 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0 ≤ (2 logb 𝑁) ↔ 1 ≤ 𝑁)) |
24 | logblt1b 47550 | . . . . 5 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℝ+) → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) | |
25 | 20, 21, 24 | syl2anc 583 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 logb 𝑁) < 1 ↔ 𝑁 < 2)) |
26 | 23, 25 | anbi12d 630 | . . 3 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (2 logb 𝑁) ∧ (2 logb 𝑁) < 1) ↔ (1 ≤ 𝑁 ∧ 𝑁 < 2))) |
27 | df-2 12291 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
28 | 27 | breq2i 5150 | . . . . . . 7 ⊢ (𝑁 < 2 ↔ 𝑁 < (1 + 1)) |
29 | 28 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 < 2 ↔ 𝑁 < (1 + 1))) |
30 | 29 | anbi2d 628 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((1 ≤ 𝑁 ∧ 𝑁 < 2) ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) |
31 | nnre 12235 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
32 | 1zzd 12609 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℤ) | |
33 | flbi 13799 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘𝑁) = 1 ↔ (1 ≤ 𝑁 ∧ 𝑁 < (1 + 1)))) | |
34 | 31, 32, 33 | syl2anc 583 | . . . . 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 12595 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
37 | flid 13791 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
38 | 36, 37 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (⌊‘𝑁) = 𝑁) |
39 | 38 | eqcomd 2733 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 = (⌊‘𝑁)) |
40 | 39 | adantr 480 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = (⌊‘𝑁)) |
41 | simpr 484 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → (⌊‘𝑁) = 1) | |
42 | 40, 41 | eqtrd 2767 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (⌊‘𝑁) = 1) → 𝑁 = 1) |
43 | 42 | ex 412 | . . . 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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 < clt 11264 ≤ cle 11265 ℕcn 12228 2c2 12283 ℤcz 12574 ℤ≥cuz 12838 ℝ+crp 12992 ⌊cfl 13773 logb clogb 26670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15651 df-ef 16029 df-sin 16031 df-cos 16032 df-pi 16034 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-limc 25769 df-dv 25770 df-log 26464 df-logb 26671 |
This theorem is referenced by: blen1b 47574 |
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