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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 12275 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ) | |
2 | 1 | nnrpd 13072 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℝ+) |
3 | nnrp 13043 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
4 | 2, 3 | rpdivcld 13091 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
5 | 4 | relogcld 26679 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ) |
6 | 5 | recnd 11286 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ) |
7 | 6 | mullidd 11276 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (1 · (log‘((𝑚 + 1) / 𝑚))) = (log‘((𝑚 + 1) / 𝑚))) |
8 | nncn 12271 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
9 | nnne0 12297 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
10 | 8, 9 | dividd 12038 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 / 𝑚) = 1) |
11 | 10 | oveq1d 7445 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 / 𝑚) + (1 / 𝑚)) = (1 + (1 / 𝑚))) |
12 | 1cnd 11253 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℂ) | |
13 | 8, 12, 8, 9 | divdird 12078 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) = ((𝑚 / 𝑚) + (1 / 𝑚))) |
14 | 8, 9 | reccld 12033 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ) |
15 | 14, 12 | addcomd 11460 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = (1 + (1 / 𝑚))) |
16 | 11, 13, 15 | 3eqtr4rd 2785 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = ((𝑚 + 1) / 𝑚)) |
17 | 16 | fveq2d 6910 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (log‘((1 / 𝑚) + 1)) = (log‘((𝑚 + 1) / 𝑚))) |
18 | 7, 17 | oveq12d 7448 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚)))) |
19 | 6 | subidd 11605 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚))) = 0) |
20 | 18, 19 | eqtrd 2774 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = 0) |
21 | 20 | mpteq2ia 5250 | . . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ 0) |
22 | fconstmpt 5750 | . . . . . 6 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) | |
23 | nnuz 12918 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
24 | 23 | xpeq1i 5714 | . . . . . 6 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0}) |
25 | 21, 22, 24 | 3eqtr2ri 2769 | . . . . 5 ⊢ ((ℤ≥‘1) × {0}) = (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) |
26 | ax-1cn 11210 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
27 | 1nn 12274 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | eldifn 4141 | . . . . . . . 8 ⊢ (1 ∈ (ℤ ∖ ℕ) → ¬ 1 ∈ ℕ) | |
29 | 27, 28 | mt2 200 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ ∖ ℕ) |
30 | eldif 3972 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ (ℤ ∖ ℕ))) | |
31 | 26, 29, 30 | mpbir2an 711 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
33 | 25, 32 | lgamcvg 27111 | . . . 4 ⊢ (⊤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1))) |
34 | 33 | mptru 1543 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1)) |
35 | log1 26641 | . . . . 5 ⊢ (log‘1) = 0 | |
36 | 35 | oveq2i 7441 | . . . 4 ⊢ ((log Γ‘1) + (log‘1)) = ((log Γ‘1) + 0) |
37 | lgamcl 27098 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘1) ∈ ℂ) | |
38 | 31, 37 | ax-mp 5 | . . . . 5 ⊢ (log Γ‘1) ∈ ℂ |
39 | 38 | addridi 11445 | . . . 4 ⊢ ((log Γ‘1) + 0) = (log Γ‘1) |
40 | 36, 39 | eqtri 2762 | . . 3 ⊢ ((log Γ‘1) + (log‘1)) = (log Γ‘1) |
41 | 34, 40 | breqtri 5172 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) |
42 | 1z 12644 | . . 3 ⊢ 1 ∈ ℤ | |
43 | serclim0 15609 | . . 3 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
44 | 42, 43 | ax-mp 5 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
45 | climuni 15584 | . 2 ⊢ ((seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) ∧ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) → (log Γ‘1) = 0) | |
46 | 41, 44, 45 | mp2an 692 | 1 ⊢ (log Γ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 ∖ cdif 3959 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 / cdiv 11917 ℕcn 12263 ℤcz 12610 ℤ≥cuz 12875 seqcseq 14038 ⇝ cli 15516 logclog 26610 log Γclgam 27073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-tan 16103 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-ulm 26434 df-log 26612 df-cxp 26613 df-lgam 27076 |
This theorem is referenced by: gam1 27122 |
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