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Theorem logfacubnd 27192

Description: A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Assertion**
Ref	Expression
logfacubnd	⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))

**Proof of Theorem logfacubnd**
Step	Hyp	Ref	Expression
1		rpre 12918	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		flge1nn 13745	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
3	1, 2	sylan 581	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
4	3	nnnn0d 12466	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 14211	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 12951	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7	6	relogcld 26592	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
9		reflcl 13720	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	8, 9	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ)
11	3	nnrpd 12951	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ⁺)
12	11	relogcld 26592	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ∈ ℝ)
13	10, 12	remulcld 11166	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ∈ ℝ)
14		relogcl 26544	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
15	14	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
16	8, 15	remulcld 11166	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · (log‘𝐴)) ∈ ℝ)
17		facubnd 14227	. . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)))
18	4, 17	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)))
19	3, 4	nnexpcld 14172	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℕ)
20	19	nnrpd 12951	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℝ⁺)
21	6, 20	logled 26596	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)) ↔ (log‘(!‘(⌊‘𝐴))) ≤ (log‘((⌊‘𝐴)↑(⌊‘𝐴)))))
22	18, 21	mpbid 232	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (log‘((⌊‘𝐴)↑(⌊‘𝐴))))
23	3	nnzd 12518	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ)
24		relogexp 26565	. . . 4 ⊢ (((⌊‘𝐴) ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℤ) → (log‘((⌊‘𝐴)↑(⌊‘𝐴))) = ((⌊‘𝐴) · (log‘(⌊‘𝐴))))
25	11, 23, 24	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘((⌊‘𝐴)↑(⌊‘𝐴))) = ((⌊‘𝐴) · (log‘(⌊‘𝐴))))
26	22, 25	breqtrd 5125	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ ((⌊‘𝐴) · (log‘(⌊‘𝐴))))
27		flle 13723	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
28	8, 27	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ≤ 𝐴)
29		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
30	11, 29	logled 26596	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ≤ 𝐴 ↔ (log‘(⌊‘𝐴)) ≤ (log‘𝐴)))
31	28, 30	mpbid 232	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ≤ (log‘𝐴))
32	11	rprege0d 12960	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ∈ ℝ ∧ 0 ≤ (⌊‘𝐴)))
33		log1 26554	. . . . . 6 ⊢ (log‘1) = 0
34	3	nnge1d 12197	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (⌊‘𝐴))
35		1rp 12913	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
36		logleb 26572	. . . . . . . 8 ⊢ ((1 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℝ⁺) → (1 ≤ (⌊‘𝐴) ↔ (log‘1) ≤ (log‘(⌊‘𝐴))))
37	35, 11, 36	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (1 ≤ (⌊‘𝐴) ↔ (log‘1) ≤ (log‘(⌊‘𝐴))))
38	34, 37	mpbid 232	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘1) ≤ (log‘(⌊‘𝐴)))
39	33, 38	eqbrtrrid 5135	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘(⌊‘𝐴)))
40	12, 39	jca 511	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((log‘(⌊‘𝐴)) ∈ ℝ ∧ 0 ≤ (log‘(⌊‘𝐴))))
41		lemul12a 12003	. . . 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 839	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((⌊‘𝐴) ≤ 𝐴 ∧ (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))))
43	28, 31, 42	mp2and 700	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))
44	7, 13, 16, 26, 43	letrd 11294	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 0cc0 11030 1c1 11031 · cmul 11035 ≤ cle 11171 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ℝ⁺crp 12909 ⌊cfl 13714 ↑cexp 13988 !cfa 14200 logclog 26523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828 df-log 26525

This theorem is referenced by: (None)

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