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Theorem gamcvg 24827

Description: The pointwise exponential of the series 𝐺 converges to Γ(𝐴) · 𝐴. (Contributed by Mario Carneiro, 6-Jul-2017.)

**Hypotheses**
Ref	Expression
lgamcvg.g	⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))
lgamcvg.a	⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))

**Assertion**
Ref	Expression
gamcvg	⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Distinct variable groups: 𝐴,𝑚 𝜑,𝑚 Allowed substitution hint: 𝐺(𝑚)

Proof of Theorem gamcvg Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11761	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11446	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		efcn 24242	. . . 4 ⊢ exp ∈ (ℂ–cn→ℂ)
4	3	a1i 11	. . 3 ⊢ (𝜑 → exp ∈ (ℂ–cn→ℂ))
5		lgamcvg.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	eldifad 3619	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
8		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
9	8	peano2nnd 11075	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
10	9	nnrpd 11908	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
11	8	nnrpd 11908	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
12	10, 11	rpdivcld 11927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
13	12	relogcld 24414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ)
14	13	recnd 10106	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ)
15	7, 14	mulcld 10098	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 · (log‘((𝑚 + 1) / 𝑚))) ∈ ℂ)
16	8	nncnd 11074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
17	8	nnne0d 11103	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
18	7, 16, 17	divcld 10839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℂ)
19		1cnd 10094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
20	18, 19	addcld 10097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℂ)
21	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
22	21, 8	dmgmdivn0 24799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0)
23	20, 22	logcld 24362	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝐴 / 𝑚) + 1)) ∈ ℂ)
24	15, 23	subcld 10430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))) ∈ ℂ)
25		lgamcvg.g	. . . . . 6 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))
26	24, 25	fmptd 6425	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ)
27	26	ffvelrnda 6399	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℂ)
28	1, 2, 27	serf 12869	. . 3 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℂ)
29	25, 5	lgamcvg 24825	. . 3 ⊢ (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
30		lgamcl 24812	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)
31	5, 30	syl 17	. . . 4 ⊢ (𝜑 → (log Γ‘𝐴) ∈ ℂ)
32	5	dmgmn0 24797	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0)
33	6, 32	logcld 24362	. . . 4 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
34	31, 33	addcld 10097	. . 3 ⊢ (𝜑 → ((log Γ‘𝐴) + (log‘𝐴)) ∈ ℂ)
35	1, 2, 4, 28, 29, 34	climcncf 22750	. 2 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ (exp‘((log Γ‘𝐴) + (log‘𝐴))))
36		efadd 14868	. . . 4 ⊢ (((log Γ‘𝐴) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
37	31, 33, 36	syl2anc 694	. . 3 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
38		eflgam 24816	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))
39	5, 38	syl 17	. . . 4 ⊢ (𝜑 → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))
40		eflog 24368	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
41	6, 32, 40	syl2anc 694	. . . 4 ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
42	39, 41	oveq12d 6708	. . 3 ⊢ (𝜑 → ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
43	37, 42	eqtrd 2685	. 2 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
44	35, 43	breqtrd 4711	1 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∖ cdif 3604 class class class wbr 4685 ↦ cmpt 4762 ∘ ccom 5147 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 / cdiv 10722 ℕcn 11058 ℤcz 11415 seqcseq 12841 ⇝ cli 14259 expce 14836 –cn→ccncf 22726 logclog 24346 log Γclgam 24787 Γcgam 24788

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-tan 14846 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-cmp 21238 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-ulm 24176 df-log 24348 df-cxp 24349 df-lgam 24790 df-gam 24791

This theorem is referenced by: gamcvg2 24831

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