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Theorem logfacubnd 27152

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 12891	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		flge1nn 13717	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
3	1, 2	sylan 580	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
4	3	nnnn0d 12434	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
5	4	faccld 14183	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℕ)
6	5	nnrpd 12924	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ∈ ℝ⁺)
7	6	relogcld 26552	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ∈ ℝ)
8	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
9		reflcl 13692	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	8, 9	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ)
11	3	nnrpd 12924	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℝ⁺)
12	11	relogcld 26552	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(⌊‘𝐴)) ∈ ℝ)
13	10, 12	remulcld 11134	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ∈ ℝ)
14		relogcl 26504	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
15	14	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
16	8, 15	remulcld 11134	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (𝐴 · (log‘𝐴)) ∈ ℝ)
17		facubnd 14199	. . . . 5 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)))
18	4, 17	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (!‘(⌊‘𝐴)) ≤ ((⌊‘𝐴)↑(⌊‘𝐴)))
19	3, 4	nnexpcld 14144	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℕ)
20	19	nnrpd 12924	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴)↑(⌊‘𝐴)) ∈ ℝ⁺)
21	6, 20	logled 26556	. . . 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 12487	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ)
24		relogexp 26525	. . . 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 5115	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ ((⌊‘𝐴) · (log‘(⌊‘𝐴))))
27		flle 13695	. . . 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 26556	. . . 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 12933	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) ∈ ℝ ∧ 0 ≤ (⌊‘𝐴)))
33		log1 26514	. . . . . 6 ⊢ (log‘1) = 0
34	3	nnge1d 12165	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → 1 ≤ (⌊‘𝐴))
35		1rp 12886	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
36		logleb 26532	. . . . . . . 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 232	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (log‘1) ≤ (log‘(⌊‘𝐴)))
39	33, 38	eqbrtrrid 5125	. . . . 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 11971	. . . 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 838	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → (((⌊‘𝐴) ≤ 𝐴 ∧ (log‘(⌊‘𝐴)) ≤ (log‘𝐴)) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))))
43	28, 31, 42	mp2and 699	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 1 ≤ 𝐴) → ((⌊‘𝐴) · (log‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴)))
44	7, 13, 16, 26, 43	letrd 11262	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 1541 ∈ wcel 2110 class class class wbr 5089 ‘cfv 6477 (class class class)co 7341 ℝcr 10997 0cc0 10998 1c1 10999 · cmul 11003 ≤ cle 11139 ℕcn 12117 ℕ₀cn0 12373 ℤcz 12460 ℝ⁺crp 12882 ⌊cfl 13686 ↑cexp 13960 !cfa 14172 logclog 26483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-submnd 18684 df-mulg 18973 df-cntz 19222 df-cmn 19687 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-xms 24228 df-ms 24229 df-tms 24230 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485

This theorem is referenced by: (None)

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