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Mirrors > Home > MPE Home > Th. List > gamcvg | Structured version Visualization version GIF version |
Description: The pointwise exponential of the series 𝐺 converges to Γ(𝐴) · 𝐴. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
lgamcvg.g | ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))) |
lgamcvg.a | ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Ref | Expression |
---|---|
gamcvg | ⊢ (𝜑 → (exp ∘ seq1( + , 𝐺)) ⇝ ((Γ‘𝐴) · 𝐴)) |
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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