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Mirrors > Home > MPE Home > Th. List > lgam1 | Structured version Visualization version GIF version |
Description: The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.) |
Ref | Expression |
---|---|
lgam1 | ⊢ (log Γ‘1) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2nn 11070 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ) | |
2 | 1 | nnrpd 11908 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℝ+) |
3 | nnrp 11880 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
4 | 2, 3 | rpdivcld 11927 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
5 | 4 | relogcld 24414 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ) |
6 | 5 | recnd 10106 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ) |
7 | 6 | mulid2d 10096 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (1 · (log‘((𝑚 + 1) / 𝑚))) = (log‘((𝑚 + 1) / 𝑚))) |
8 | nncn 11066 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
9 | nnne0 11091 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
10 | 8, 9 | dividd 10837 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 / 𝑚) = 1) |
11 | 10 | oveq1d 6705 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 / 𝑚) + (1 / 𝑚)) = (1 + (1 / 𝑚))) |
12 | 1cnd 10094 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℂ) | |
13 | 8, 12, 8, 9 | divdird 10877 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) = ((𝑚 / 𝑚) + (1 / 𝑚))) |
14 | 8, 9 | reccld 10832 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ) |
15 | 14, 12 | addcomd 10276 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = (1 + (1 / 𝑚))) |
16 | 11, 13, 15 | 3eqtr4rd 2696 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = ((𝑚 + 1) / 𝑚)) |
17 | 16 | fveq2d 6233 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (log‘((1 / 𝑚) + 1)) = (log‘((𝑚 + 1) / 𝑚))) |
18 | 7, 17 | oveq12d 6708 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚)))) |
19 | 6 | subidd 10418 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚))) = 0) |
20 | 18, 19 | eqtrd 2685 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = 0) |
21 | 20 | mpteq2ia 4773 | . . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ 0) |
22 | fconstmpt 5197 | . . . . . 6 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) | |
23 | nnuz 11761 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
24 | 23 | xpeq1i 5169 | . . . . . 6 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0}) |
25 | 21, 22, 24 | 3eqtr2ri 2680 | . . . . 5 ⊢ ((ℤ≥‘1) × {0}) = (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) |
26 | ax-1cn 10032 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
27 | 1nn 11069 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | eldifn 3766 | . . . . . . . 8 ⊢ (1 ∈ (ℤ ∖ ℕ) → ¬ 1 ∈ ℕ) | |
29 | 27, 28 | mt2 191 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ ∖ ℕ) |
30 | eldif 3617 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ (ℤ ∖ ℕ))) | |
31 | 26, 29, 30 | mpbir2an 975 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
33 | 25, 32 | lgamcvg 24825 | . . . 4 ⊢ (⊤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1))) |
34 | 33 | trud 1533 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1)) |
35 | log1 24377 | . . . . 5 ⊢ (log‘1) = 0 | |
36 | 35 | oveq2i 6701 | . . . 4 ⊢ ((log Γ‘1) + (log‘1)) = ((log Γ‘1) + 0) |
37 | lgamcl 24812 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘1) ∈ ℂ) | |
38 | 31, 37 | ax-mp 5 | . . . . 5 ⊢ (log Γ‘1) ∈ ℂ |
39 | 38 | addid1i 10261 | . . . 4 ⊢ ((log Γ‘1) + 0) = (log Γ‘1) |
40 | 36, 39 | eqtri 2673 | . . 3 ⊢ ((log Γ‘1) + (log‘1)) = (log Γ‘1) |
41 | 34, 40 | breqtri 4710 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) |
42 | 1z 11445 | . . 3 ⊢ 1 ∈ ℤ | |
43 | serclim0 14352 | . . 3 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
44 | 42, 43 | ax-mp 5 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
45 | climuni 14327 | . 2 ⊢ ((seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) ∧ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) → (log Γ‘1) = 0) | |
46 | 41, 44, 45 | mp2an 708 | 1 ⊢ (log Γ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ∖ cdif 3604 {csn 4210 class class class wbr 4685 ↦ cmpt 4762 × cxp 5141 ‘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 ℤ≥cuz 11725 seqcseq 12841 ⇝ cli 14259 logclog 24346 log Γclgam 24787 |
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 |
This theorem is referenced by: gam1 24836 |
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