relgamcl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > relgamcl		Structured version Visualization version GIF version

Theorem relgamcl 25331

Description: The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)

**Assertion**
Ref	Expression
relgamcl	⊢ (𝐴 ∈ ℝ⁺ → (log Γ‘𝐴) ∈ ℝ)

Proof of Theorem relgamcl Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpdmgm 25294	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
2		lgamcl 25310	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (log Γ‘𝐴) ∈ ℂ)
4		relogcl 24850	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 10460	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6	3, 5	pncand 10791	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((log Γ‘𝐴) + (log‘𝐴)) − (log‘𝐴)) = (log Γ‘𝐴))
7		nnuz 12088	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
8		1zzd 11819	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℤ)
9		eqid 2772	. . . . 5 ⊢ (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))
10	9, 1	lgamcvg 25323	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
11		simpl 475	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
12	11	rpred 12241	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℝ)
13		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
14	13	peano2nnd 11450	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
15	14	nnrpd 12239	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
16	13	nnrpd 12239	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
17	15, 16	rpdivcld 12258	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
18	17	relogcld 24897	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ)
19	12, 18	remulcld 10462	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (𝐴 · (log‘((𝑚 + 1) / 𝑚))) ∈ ℝ)
20	11, 16	rpdivcld 12258	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℝ⁺)
21		1rp 12201	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
22	21	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ⁺)
23	20, 22	rpaddcld 12256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℝ⁺)
24	23	relogcld 24897	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → (log‘((𝐴 / 𝑚) + 1)) ∈ ℝ)
25	19, 24	resubcld 10861	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℕ) → ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))) ∈ ℝ)
26	25	fmpttd 6696	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))):ℕ⟶ℝ)
27	26	ffvelrnda 6670	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))‘𝑛) ∈ ℝ)
28	7, 8, 27	serfre 13207	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))):ℕ⟶ℝ)
29	28	ffvelrnda 6670	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℕ) → (seq1( + , (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))))‘𝑛) ∈ ℝ)
30	7, 8, 10, 29	climrecl 14791	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((log Γ‘𝐴) + (log‘𝐴)) ∈ ℝ)
31	30, 4	resubcld 10861	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((log Γ‘𝐴) + (log‘𝐴)) − (log‘𝐴)) ∈ ℝ)
32	6, 31	eqeltrrd 2861	1 ⊢ (𝐴 ∈ ℝ⁺ → (log Γ‘𝐴) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2048 ∖ cdif 3822 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 ℝcr 10326 1c1 10328 + caddc 10330 · cmul 10332 − cmin 10662 / cdiv 11090 ℕcn 11431 ℤcz 11786 ℝ⁺crp 12197 seqcseq 13177 logclog 24829 log Γclgam 25285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-dju 9116 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-fac 13442 df-bc 13471 df-hash 13499 df-shft 14277 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-limsup 14679 df-clim 14696 df-rlim 14697 df-sum 14894 df-ef 15271 df-sin 15273 df-cos 15274 df-tan 15275 df-pi 15276 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-lp 21438 df-perf 21439 df-cn 21529 df-cnp 21530 df-haus 21617 df-cmp 21689 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-xms 22623 df-ms 22624 df-tms 22625 df-cncf 23179 df-limc 24157 df-dv 24158 df-ulm 24658 df-log 24831 df-cxp 24832 df-lgam 25288

This theorem is referenced by: rpgamcl 25332

W3C validator