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Mirrors > Home > MPE Home > Th. List > logfacubnd | Structured version Visualization version GIF version |
Description: A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.) |
Ref | Expression |
---|---|
logfacubnd | ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12247 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | flge1nn 13041 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ) | |
3 | 1, 2 | sylan 580 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ) |
4 | 3 | nnnn0d 11803 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) |
5 | 4 | faccld 13494 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ) |
6 | 5 | nnrpd 12279 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ+) |
7 | 6 | relogcld 24887 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ) |
8 | 1 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ) |
9 | reflcl 13016 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ) |
11 | 3 | nnrpd 12279 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ+) |
12 | 11 | relogcld 24887 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ∈ ℝ) |
13 | 10, 12 | remulcld 10517 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ∈ ℝ) |
14 | relogcl 24840 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
15 | 14 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ) |
16 | 8, 15 | remulcld 10517 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (𝐴 · (log‘𝐴)) ∈ ℝ) |
17 | facubnd 13510 | . . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ0 → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴))) | |
18 | 4, 17 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴))) |
19 | 3, 4 | nnexpcld 13456 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℕ) |
20 | 19 | nnrpd 12279 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℝ+) |
21 | 6, 20 | logled 24891 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)) ↔ (log‘(!‘(⌊‘𝐴))) ≤ (log‘((⌊‘𝐴)↑(⌊‘𝐴))))) |
22 | 18, 21 | mpbid 233 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (log‘((⌊‘𝐴)↑(⌊‘𝐴)))) |
23 | 3 | nnzd 11935 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ) |
24 | relogexp 24860 | . . . 4 ⊢ (((⌊‘𝐴) ∈ ℝ+ ∧ (⌊‘𝐴) ∈ ℤ) → (log‘((⌊‘𝐴)↑(⌊‘𝐴))) = ((⌊‘𝐴) · (log‘(⌊‘𝐴)))) | |
25 | 11, 23, 24 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘((⌊‘𝐴)↑(⌊‘𝐴))) = ((⌊‘𝐴) · (log‘(⌊‘𝐴)))) |
26 | 22, 25 | breqtrd 4988 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ ((⌊‘𝐴) · (log‘(⌊‘𝐴)))) |
27 | flle 13019 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
28 | 8, 27 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ≤ 𝐴) |
29 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ+) | |
30 | 11, 29 | logled 24891 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ≤ 𝐴 ↔ (log‘(⌊‘𝐴)) ≤ (log‘𝐴))) |
31 | 28, 30 | mpbid 233 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) |
32 | 11 | rprege0d 12288 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ∈ ℝ ∧ 0 ≤ (⌊‘𝐴))) |
33 | log1 24850 | . . . . . 6 ⊢ (log‘1) = 0 | |
34 | 3 | nnge1d 11533 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 1 ≤ (⌊‘𝐴)) |
35 | 1rp 12243 | . . . . . . . 8 ⊢ 1 ∈ ℝ+ | |
36 | logleb 24867 | . . . . . . . 8 ⊢ ((1 ∈ ℝ+ ∧ (⌊‘𝐴) ∈ ℝ+) → (1 ≤ (⌊‘𝐴) ↔ (log‘1) ≤ (log‘(⌊‘𝐴)))) | |
37 | 35, 11, 36 | sylancr 587 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (1 ≤ (⌊‘𝐴) ↔ (log‘1) ≤ (log‘(⌊‘𝐴)))) |
38 | 34, 37 | mpbid 233 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘1) ≤ (log‘(⌊‘𝐴))) |
39 | 33, 38 | eqbrtrrid 4998 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘(⌊‘𝐴))) |
40 | 12, 39 | jca 512 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((log‘(⌊‘𝐴)) ∈ ℝ ∧ 0 ≤ (log‘(⌊‘𝐴)))) |
41 | lemul12a 11346 | . . . 4 ⊢ (((((⌊‘𝐴) ∈ ℝ ∧ 0 ≤ (⌊‘𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (((log‘(⌊‘𝐴)) ∈ ℝ ∧ 0 ≤ (log‘(⌊‘𝐴))) ∧ (log‘𝐴) ∈ ℝ)) → (((⌊‘𝐴) ≤ 𝐴 ∧ (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))) | |
42 | 32, 8, 40, 15, 41 | syl22anc 835 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (((⌊‘𝐴) ≤ 𝐴 ∧ (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))) |
43 | 28, 31, 42 | mp2and 695 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))) |
44 | 7, 13, 16, 26, 43 | letrd 10644 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 · cmul 10388 ≤ cle 10522 ℕcn 11486 ℕ0cn0 11745 ℤcz 11829 ℝ+crp 12239 ⌊cfl 13010 ↑cexp 13279 !cfa 13483 logclog 24819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-limc 24147 df-dv 24148 df-log 24821 |
This theorem is referenced by: (None) |
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