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Theorem gamcvg 24527

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 11558	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11244	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		efcn 23946	. . . 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 3552	. . . . . . . . 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 10887	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
10	9	nnrpd 11705	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
11	8	nnrpd 11705	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
12	10, 11	rpdivcld 11724	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
13	12	relogcld 24118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ)
14	13	recnd 9925	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ)
15	7, 14	mulcld 9917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 · (log‘((𝑚 + 1) / 𝑚))) ∈ ℂ)
16	8	nncnd 10886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
17	8	nnne0d 10915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
18	7, 16, 17	divcld 10653	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℂ)
19		1cnd 9913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
20	18, 19	addcld 9916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℂ)
21	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
22	21, 8	dmgmdivn0 24499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0)
23	20, 22	logcld 24066	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝐴 / 𝑚) + 1)) ∈ ℂ)
24	15, 23	subcld 10244	. . . . . 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 6277	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ)
27	26	ffvelrnda 6252	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℂ)
28	1, 2, 27	serf 12649	. . 3 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℂ)
29	25, 5	lgamcvg 24525	. . 3 ⊢ (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
30		lgamcl 24512	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)
31	5, 30	syl 17	. . . 4 ⊢ (𝜑 → (log Γ‘𝐴) ∈ ℂ)
32	5	dmgmn0 24497	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0)
33	6, 32	logcld 24066	. . . 4 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
34	31, 33	addcld 9916	. . 3 ⊢ (𝜑 → ((log Γ‘𝐴) + (log‘𝐴)) ∈ ℂ)
35	1, 2, 4, 28, 29, 34	climcncf 22459	. 2 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ (exp‘((log Γ‘𝐴) + (log‘𝐴))))
36		efadd 14612	. . . 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 691	. . 3 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
38		eflgam 24516	. . . . 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 24072	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
41	6, 32, 40	syl2anc 691	. . . 4 ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
42	39, 41	oveq12d 6545	. . 3 ⊢ (𝜑 → ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
43	37, 42	eqtrd 2644	. 2 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
44	35, 43	breqtrd 4604	1 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 class class class wbr 4578 ↦ cmpt 4638 ∘ ccom 5032 ‘cfv 5790 (class class class)co 6527 ℂcc 9791 0cc0 9793 1c1 9794 + caddc 9796 · cmul 9798 − cmin 10118 / cdiv 10536 ℕcn 10870 ℤcz 11213 seqcseq 12621 ⇝ cli 14012 expce 14580 –cn→ccncf 22435 logclog 24050 log Γclgam 24487 Γcgam 24488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-inf2 8399 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870 ax-pre-sup 9871 ax-addf 9872 ax-mulf 9873

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4368 df-int 4406 df-iun 4452 df-iin 4453 df-br 4579 df-opab 4639 df-mpt 4640 df-tr 4676 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-se 4988 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-isom 5799 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-of 6773 df-om 6936 df-1st 7037 df-2nd 7038 df-supp 7161 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-2o 7426 df-oadd 7429 df-er 7607 df-map 7724 df-pm 7725 df-ixp 7773 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-fsupp 8137 df-fi 8178 df-sup 8209 df-inf 8210 df-oi 8276 df-card 8626 df-cda 8851 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-div 10537 df-nn 10871 df-2 10929 df-3 10930 df-4 10931 df-5 10932 df-6 10933 df-7 10934 df-8 10935 df-9 10936 df-n0 11143 df-z 11214 df-dec 11329 df-uz 11523 df-q 11624 df-rp 11668 df-xneg 11781 df-xadd 11782 df-xmul 11783 df-ioo 12009 df-ioc 12010 df-ico 12011 df-icc 12012 df-fz 12156 df-fzo 12293 df-fl 12413 df-mod 12489 df-seq 12622 df-exp 12681 df-fac 12881 df-bc 12910 df-hash 12938 df-shft 13604 df-cj 13636 df-re 13637 df-im 13638 df-sqrt 13772 df-abs 13773 df-limsup 13999 df-clim 14016 df-rlim 14017 df-sum 14214 df-ef 14586 df-sin 14588 df-cos 14589 df-tan 14590 df-pi 14591 df-struct 15646 df-ndx 15647 df-slot 15648 df-base 15649 df-sets 15650 df-ress 15651 df-plusg 15730 df-mulr 15731 df-starv 15732 df-sca 15733 df-vsca 15734 df-ip 15735 df-tset 15736 df-ple 15737 df-ds 15740 df-unif 15741 df-hom 15742 df-cco 15743 df-rest 15855 df-topn 15856 df-0g 15874 df-gsum 15875 df-topgen 15876 df-pt 15877 df-prds 15880 df-xrs 15934 df-qtop 15939 df-imas 15940 df-xps 15942 df-mre 16018 df-mrc 16019 df-acs 16021 df-mgm 17014 df-sgrp 17056 df-mnd 17067 df-submnd 17108 df-mulg 17313 df-cntz 17522 df-cmn 17967 df-psmet 19508 df-xmet 19509 df-met 19510 df-bl 19511 df-mopn 19512 df-fbas 19513 df-fg 19514 df-cnfld 19517 df-top 20469 df-bases 20470 df-topon 20471 df-topsp 20472 df-cld 20581 df-ntr 20582 df-cls 20583 df-nei 20660 df-lp 20698 df-perf 20699 df-cn 20789 df-cnp 20790 df-haus 20877 df-cmp 20948 df-tx 21123 df-hmeo 21316 df-fil 21408 df-fm 21500 df-flim 21501 df-flf 21502 df-xms 21883 df-ms 21884 df-tms 21885 df-cncf 22437 df-limc 23381 df-dv 23382 df-ulm 23880 df-log 24052 df-cxp 24053 df-lgam 24490 df-gam 24491

This theorem is referenced by: gamcvg2 24531

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