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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fllogbd | Structured version Visualization version GIF version |
Description: A real number is between the base of a logarithm to the power of the floor of the logarithm of the number and the base of the logarithm to the power of the floor of the logarithm of the number plus one. (Contributed by AV, 23-May-2020.) |
Ref | Expression |
---|---|
fllogbd.b | ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) |
fllogbd.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
fllogbd.e | ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) |
Ref | Expression |
---|---|
fllogbd | ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fllogbd.e | . . . . 5 ⊢ 𝐸 = (⌊‘(𝐵 logb 𝑋)) | |
2 | fllogbd.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) | |
3 | fllogbd.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
4 | relogbzcl 26726 | . . . . . . 7 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℝ) |
6 | flle 13804 | . . . . . 6 ⊢ ((𝐵 logb 𝑋) ∈ ℝ → (⌊‘(𝐵 logb 𝑋)) ≤ (𝐵 logb 𝑋)) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐵 logb 𝑋)) ≤ (𝐵 logb 𝑋)) |
8 | 1, 7 | eqbrtrid 5187 | . . . 4 ⊢ (𝜑 → 𝐸 ≤ (𝐵 logb 𝑋)) |
9 | eluzelz 12870 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℤ) | |
10 | 2, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
11 | 10 | zred 12704 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
12 | eluz2b1 12941 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) ↔ (𝐵 ∈ ℤ ∧ 1 < 𝐵)) | |
13 | 12 | simprbi 495 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) |
14 | 2, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 < 𝐵) |
15 | 5 | flcld 13803 | . . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐵 logb 𝑋)) ∈ ℤ) |
16 | 1, 15 | eqeltrid 2833 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ) |
17 | 16 | zred 12704 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
18 | 11, 14, 17, 5 | cxpled 26674 | . . . 4 ⊢ (𝜑 → (𝐸 ≤ (𝐵 logb 𝑋) ↔ (𝐵↑𝑐𝐸) ≤ (𝐵↑𝑐(𝐵 logb 𝑋)))) |
19 | 8, 18 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐𝐸) ≤ (𝐵↑𝑐(𝐵 logb 𝑋))) |
20 | 10 | zcnd 12705 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | eluz2nn 12906 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ) | |
22 | 2, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
23 | 22 | nnne0d 12300 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 0) |
24 | 20, 23, 16 | cxpexpzd 26665 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐𝐸) = (𝐵↑𝐸)) |
25 | eluz2cnn0n1 47657 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ (ℂ ∖ {0, 1})) | |
26 | 2, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ {0, 1})) |
27 | rpcnne0 13032 | . . . . . 6 ⊢ (𝑋 ∈ ℝ+ → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
28 | eldifsn 4795 | . . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
29 | 27, 28 | sylibr 233 | . . . . 5 ⊢ (𝑋 ∈ ℝ+ → 𝑋 ∈ (ℂ ∖ {0})) |
30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (ℂ ∖ {0})) |
31 | cxplogb 26738 | . . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) | |
32 | 26, 30, 31 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
33 | 19, 24, 32 | 3brtr3d 5183 | . 2 ⊢ (𝜑 → (𝐵↑𝐸) ≤ 𝑋) |
34 | flltp1 13805 | . . . . . 6 ⊢ ((𝐵 logb 𝑋) ∈ ℝ → (𝐵 logb 𝑋) < ((⌊‘(𝐵 logb 𝑋)) + 1)) | |
35 | 5, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐵 logb 𝑋) < ((⌊‘(𝐵 logb 𝑋)) + 1)) |
36 | 1 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐸 = (⌊‘(𝐵 logb 𝑋))) |
37 | 36 | oveq1d 7441 | . . . . 5 ⊢ (𝜑 → (𝐸 + 1) = ((⌊‘(𝐵 logb 𝑋)) + 1)) |
38 | 35, 37 | breqtrrd 5180 | . . . 4 ⊢ (𝜑 → (𝐵 logb 𝑋) < (𝐸 + 1)) |
39 | 16 | peano2zd 12707 | . . . . . 6 ⊢ (𝜑 → (𝐸 + 1) ∈ ℤ) |
40 | 39 | zred 12704 | . . . . 5 ⊢ (𝜑 → (𝐸 + 1) ∈ ℝ) |
41 | 11, 14, 5, 40 | cxpltd 26673 | . . . 4 ⊢ (𝜑 → ((𝐵 logb 𝑋) < (𝐸 + 1) ↔ (𝐵↑𝑐(𝐵 logb 𝑋)) < (𝐵↑𝑐(𝐸 + 1)))) |
42 | 38, 41 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) < (𝐵↑𝑐(𝐸 + 1))) |
43 | 20, 23, 39 | cxpexpzd 26665 | . . 3 ⊢ (𝜑 → (𝐵↑𝑐(𝐸 + 1)) = (𝐵↑(𝐸 + 1))) |
44 | 42, 32, 43 | 3brtr3d 5183 | . 2 ⊢ (𝜑 → 𝑋 < (𝐵↑(𝐸 + 1))) |
45 | 33, 44 | jca 510 | 1 ⊢ (𝜑 → ((𝐵↑𝐸) ≤ 𝑋 ∧ 𝑋 < (𝐵↑(𝐸 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∖ cdif 3946 {csn 4632 {cpr 4634 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 ℕcn 12250 2c2 12305 ℤcz 12596 ℤ≥cuz 12860 ℝ+crp 13014 ⌊cfl 13795 ↑cexp 14066 ↑𝑐ccxp 26509 logb clogb 26716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-cxp 26511 df-logb 26717 |
This theorem is referenced by: fldivexpfllog2 47716 |
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