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Theorem gamcvg 27097

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 12875	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12599	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		efcn 26483	. . . 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 3916	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
7	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
8		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
9	8	peano2nnd 12224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
10	9	nnrpd 13032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
11	8	nnrpd 13032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
12	10, 11	rpdivcld 13051	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
13	12	relogcld 26665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ)
14	13	recnd 11207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ)
15	7, 14	mulcld 11199	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 · (log‘((𝑚 + 1) / 𝑚))) ∈ ℂ)
16	8	nncnd 12223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
17	8	nnne0d 12260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
18	7, 16, 17	divcld 11964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℂ)
19		1cnd 11172	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℂ)
20	18, 19	addcld 11198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℂ)
21	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
22	21, 8	dmgmdivn0 27069	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0)
23	20, 22	logcld 26612	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘((𝐴 / 𝑚) + 1)) ∈ ℂ)
24	15, 23	subcld 11539	. . . . . 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 7091	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶ℂ)
27	26	ffvelcdmda 7061	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℂ)
28	1, 2, 27	serf 14040	. . 3 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℂ)
29	25, 5	lgamcvg 27095	. . 3 ⊢ (𝜑 → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
30		lgamcl 27082	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)
31	5, 30	syl 17	. . . 4 ⊢ (𝜑 → (log Γ‘𝐴) ∈ ℂ)
32	5	dmgmn0 27067	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0)
33	6, 32	logcld 26612	. . . 4 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
34	31, 33	addcld 11198	. . 3 ⊢ (𝜑 → ((log Γ‘𝐴) + (log‘𝐴)) ∈ ℂ)
35	1, 2, 4, 28, 29, 34	climcncf 24942	. 2 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ (exp‘((log Γ‘𝐴) + (log‘𝐴))))
36		efadd 16107	. . . 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 593	. . 3 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
38		eflgam 27086	. . . . 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 26618	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
41	6, 32, 40	syl2anc 593	. . . 4 ⊢ (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
42	39, 41	oveq12d 7410	. . 3 ⊢ (𝜑 → ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
43	37, 42	eqtrd 2796	. 2 ⊢ (𝜑 → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((Γ‘𝐴) · 𝐴))
44	35, 43	breqtrd 5125	1 ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3901 class class class wbr 5099 ↦ cmpt 5180 ∘ ccom 5649 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 − cmin 11411 / cdiv 11841 ℕcn 12207 ℤcz 12565 seqcseq 14011 ⇝ cli 15494 expce 16074 –cn→ccncf 24918 logclog 26596 log Γclgam 27057 Γcgam 27058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-dju 9856 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-tan 16084 df-pi 16085 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-haus 23355 df-cmp 23427 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-ulm 26417 df-log 26598 df-cxp 26599 df-lgam 27060 df-gam 27061

This theorem is referenced by: gamcvg2 27101

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