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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 12257 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ) | |
2 | 1 | nnrpd 13049 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℝ+) |
3 | nnrp 13020 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
4 | 2, 3 | rpdivcld 13068 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
5 | 4 | relogcld 26602 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ) |
6 | 5 | recnd 11274 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ) |
7 | 6 | mullidd 11264 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (1 · (log‘((𝑚 + 1) / 𝑚))) = (log‘((𝑚 + 1) / 𝑚))) |
8 | nncn 12253 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
9 | nnne0 12279 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
10 | 8, 9 | dividd 12021 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 / 𝑚) = 1) |
11 | 10 | oveq1d 7434 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 / 𝑚) + (1 / 𝑚)) = (1 + (1 / 𝑚))) |
12 | 1cnd 11241 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℂ) | |
13 | 8, 12, 8, 9 | divdird 12061 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) = ((𝑚 / 𝑚) + (1 / 𝑚))) |
14 | 8, 9 | reccld 12016 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ) |
15 | 14, 12 | addcomd 11448 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = (1 + (1 / 𝑚))) |
16 | 11, 13, 15 | 3eqtr4rd 2776 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ((1 / 𝑚) + 1) = ((𝑚 + 1) / 𝑚)) |
17 | 16 | fveq2d 6900 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (log‘((1 / 𝑚) + 1)) = (log‘((𝑚 + 1) / 𝑚))) |
18 | 7, 17 | oveq12d 7437 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚)))) |
19 | 6 | subidd 11591 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚))) = 0) |
20 | 18, 19 | eqtrd 2765 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = 0) |
21 | 20 | mpteq2ia 5252 | . . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ 0) |
22 | fconstmpt 5740 | . . . . . 6 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) | |
23 | nnuz 12898 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
24 | 23 | xpeq1i 5704 | . . . . . 6 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0}) |
25 | 21, 22, 24 | 3eqtr2ri 2760 | . . . . 5 ⊢ ((ℤ≥‘1) × {0}) = (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) |
26 | ax-1cn 11198 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
27 | 1nn 12256 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | eldifn 4124 | . . . . . . . 8 ⊢ (1 ∈ (ℤ ∖ ℕ) → ¬ 1 ∈ ℕ) | |
29 | 27, 28 | mt2 199 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ ∖ ℕ) |
30 | eldif 3954 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (1 ∈ ℂ ∧ ¬ 1 ∈ (ℤ ∖ ℕ))) | |
31 | 26, 29, 30 | mpbir2an 709 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
33 | 25, 32 | lgamcvg 27031 | . . . 4 ⊢ (⊤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1))) |
34 | 33 | mptru 1540 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1)) |
35 | log1 26564 | . . . . 5 ⊢ (log‘1) = 0 | |
36 | 35 | oveq2i 7430 | . . . 4 ⊢ ((log Γ‘1) + (log‘1)) = ((log Γ‘1) + 0) |
37 | lgamcl 27018 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘1) ∈ ℂ) | |
38 | 31, 37 | ax-mp 5 | . . . . 5 ⊢ (log Γ‘1) ∈ ℂ |
39 | 38 | addridi 11433 | . . . 4 ⊢ ((log Γ‘1) + 0) = (log Γ‘1) |
40 | 36, 39 | eqtri 2753 | . . 3 ⊢ ((log Γ‘1) + (log‘1)) = (log Γ‘1) |
41 | 34, 40 | breqtri 5174 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) |
42 | 1z 12625 | . . 3 ⊢ 1 ∈ ℤ | |
43 | serclim0 15557 | . . 3 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
44 | 42, 43 | ax-mp 5 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
45 | climuni 15532 | . 2 ⊢ ((seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) ∧ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) → (log Γ‘1) = 0) | |
46 | 41, 44, 45 | mp2an 690 | 1 ⊢ (log Γ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∖ cdif 3941 {csn 4630 class class class wbr 5149 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 − cmin 11476 / cdiv 11903 ℕcn 12245 ℤcz 12591 ℤ≥cuz 12855 seqcseq 14002 ⇝ cli 15464 logclog 26533 log Γclgam 26993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-tan 16051 df-pi 16052 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 df-ulm 26358 df-log 26535 df-cxp 26536 df-lgam 26996 |
This theorem is referenced by: gam1 27042 |
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