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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 11985 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ) | |
2 | 1 | nnrpd 12770 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℝ+) |
3 | nnrp 12741 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
4 | 2, 3 | rpdivcld 12789 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
5 | 4 | relogcld 25778 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ) |
6 | 5 | recnd 11003 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ) |
7 | 6 | mulid2d 10993 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (1 · (log‘((𝑚 + 1) / 𝑚))) = (log‘((𝑚 + 1) / 𝑚))) |
8 | nncn 11981 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
9 | nnne0 12007 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
10 | 8, 9 | dividd 11749 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 / 𝑚) = 1) |
11 | 10 | oveq1d 7290 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 / 𝑚) + (1 / 𝑚)) = (1 + (1 / 𝑚))) |
12 | 1cnd 10970 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℂ) | |
13 | 8, 12, 8, 9 | divdird 11789 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) = ((𝑚 / 𝑚) + (1 / 𝑚))) |
14 | 8, 9 | reccld 11744 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ) |
15 | 14, 12 | addcomd 11177 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = (1 + (1 / 𝑚))) |
16 | 11, 13, 15 | 3eqtr4rd 2789 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = ((𝑚 + 1) / 𝑚)) |
17 | 16 | fveq2d 6778 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (log‘((1 / 𝑚) + 1)) = (log‘((𝑚 + 1) / 𝑚))) |
18 | 7, 17 | oveq12d 7293 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚)))) |
19 | 6 | subidd 11320 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚))) = 0) |
20 | 18, 19 | eqtrd 2778 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = 0) |
21 | 20 | mpteq2ia 5177 | . . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ 0) |
22 | fconstmpt 5649 | . . . . . 6 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) | |
23 | nnuz 12621 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
24 | 23 | xpeq1i 5615 | . . . . . 6 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0}) |
25 | 21, 22, 24 | 3eqtr2ri 2773 | . . . . 5 ⊢ ((ℤ≥‘1) × {0}) = (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) |
26 | ax-1cn 10929 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
27 | 1nn 11984 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | eldifn 4062 | . . . . . . . 8 ⊢ (1 ∈ (ℤ ∖ ℕ) → ¬ 1 ∈ ℕ) | |
29 | 27, 28 | mt2 199 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ ∖ ℕ) |
30 | eldif 3897 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ (ℤ ∖ ℕ))) | |
31 | 26, 29, 30 | mpbir2an 708 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
33 | 25, 32 | lgamcvg 26203 | . . . 4 ⊢ (⊤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1))) |
34 | 33 | mptru 1546 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1)) |
35 | log1 25741 | . . . . 5 ⊢ (log‘1) = 0 | |
36 | 35 | oveq2i 7286 | . . . 4 ⊢ ((log Γ‘1) + (log‘1)) = ((log Γ‘1) + 0) |
37 | lgamcl 26190 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘1) ∈ ℂ) | |
38 | 31, 37 | ax-mp 5 | . . . . 5 ⊢ (log Γ‘1) ∈ ℂ |
39 | 38 | addid1i 11162 | . . . 4 ⊢ ((log Γ‘1) + 0) = (log Γ‘1) |
40 | 36, 39 | eqtri 2766 | . . 3 ⊢ ((log Γ‘1) + (log‘1)) = (log Γ‘1) |
41 | 34, 40 | breqtri 5099 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) |
42 | 1z 12350 | . . 3 ⊢ 1 ∈ ℤ | |
43 | serclim0 15286 | . . 3 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
44 | 42, 43 | ax-mp 5 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
45 | climuni 15261 | . 2 ⊢ ((seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) ∧ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) → (log Γ‘1) = 0) | |
46 | 41, 44, 45 | mp2an 689 | 1 ⊢ (log Γ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ∖ cdif 3884 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 − cmin 11205 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℤ≥cuz 12582 seqcseq 13721 ⇝ cli 15193 logclog 25710 log Γclgam 26165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536 df-log 25712 df-cxp 25713 df-lgam 26168 |
This theorem is referenced by: gam1 26214 |
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