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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 11740 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ) | |
2 | 1 | nnrpd 12524 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℝ+) |
3 | nnrp 12495 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+) | |
4 | 2, 3 | rpdivcld 12543 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
5 | 4 | relogcld 25378 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℝ) |
6 | 5 | recnd 10759 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (log‘((𝑚 + 1) / 𝑚)) ∈ ℂ) |
7 | 6 | mulid2d 10749 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (1 · (log‘((𝑚 + 1) / 𝑚))) = (log‘((𝑚 + 1) / 𝑚))) |
8 | nncn 11736 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
9 | nnne0 11762 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
10 | 8, 9 | dividd 11504 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (𝑚 / 𝑚) = 1) |
11 | 10 | oveq1d 7197 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 / 𝑚) + (1 / 𝑚)) = (1 + (1 / 𝑚))) |
12 | 1cnd 10726 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℂ) | |
13 | 8, 12, 8, 9 | divdird 11544 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ((𝑚 + 1) / 𝑚) = ((𝑚 / 𝑚) + (1 / 𝑚))) |
14 | 8, 9 | reccld 11499 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (1 / 𝑚) ∈ ℂ) |
15 | 14, 12 | addcomd 10932 | . . . . . . . . . . 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 6690 | . . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (log‘((1 / 𝑚) + 1)) = (log‘((𝑚 + 1) / 𝑚))) |
18 | 7, 17 | oveq12d 7200 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1))) = ((log‘((𝑚 + 1) / 𝑚)) − (log‘((𝑚 + 1) / 𝑚)))) |
19 | 6 | subidd 11075 | . . . . . . . 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 5131 | . . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ((1 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((1 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ 0) |
22 | fconstmpt 5595 | . . . . . 6 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) | |
23 | nnuz 12375 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
24 | 23 | xpeq1i 5561 | . . . . . 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 10685 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
27 | 1nn 11739 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
28 | eldifn 4028 | . . . . . . . 8 ⊢ (1 ∈ (ℤ ∖ ℕ) → ¬ 1 ∈ ℕ) | |
29 | 27, 28 | mt2 203 | . . . . . . 7 ⊢ ¬ 1 ∈ (ℤ ∖ ℕ) |
30 | eldif 3863 | . . . . . . 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 25803 | . . . 4 ⊢ (⊤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1))) |
34 | 33 | mptru 1549 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ ((log Γ‘1) + (log‘1)) |
35 | log1 25341 | . . . . 5 ⊢ (log‘1) = 0 | |
36 | 35 | oveq2i 7193 | . . . 4 ⊢ ((log Γ‘1) + (log‘1)) = ((log Γ‘1) + 0) |
37 | lgamcl 25790 | . . . . . 6 ⊢ (1 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘1) ∈ ℂ) | |
38 | 31, 37 | ax-mp 5 | . . . . 5 ⊢ (log Γ‘1) ∈ ℂ |
39 | 38 | addid1i 10917 | . . . 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 5065 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ (log Γ‘1) |
42 | 1z 12105 | . . 3 ⊢ 1 ∈ ℤ | |
43 | serclim0 15036 | . . 3 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
44 | 42, 43 | ax-mp 5 | . 2 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
45 | climuni 15011 | . 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 1542 ⊤wtru 1543 ∈ wcel 2114 ∖ cdif 3850 {csn 4526 class class class wbr 5040 ↦ cmpt 5120 × cxp 5533 ‘cfv 6349 (class class class)co 7182 ℂcc 10625 0cc0 10627 1c1 10628 + caddc 10630 · cmul 10632 − cmin 10960 / cdiv 11387 ℕcn 11728 ℤcz 12074 ℤ≥cuz 12336 seqcseq 13472 ⇝ cli 14943 logclog 25310 log Γclgam 25765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-oadd 8147 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-dju 9415 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ioc 12838 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-fac 13738 df-bc 13767 df-hash 13795 df-shft 14528 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-limsup 14930 df-clim 14947 df-rlim 14948 df-sum 15148 df-ef 15525 df-sin 15527 df-cos 15528 df-tan 15529 df-pi 15530 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-hom 16704 df-cco 16705 df-rest 16811 df-topn 16812 df-0g 16830 df-gsum 16831 df-topgen 16832 df-pt 16833 df-prds 16836 df-xrs 16890 df-qtop 16895 df-imas 16896 df-xps 16898 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-mulg 18355 df-cntz 18577 df-cmn 19038 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-fbas 20226 df-fg 20227 df-cnfld 20230 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-cld 21782 df-ntr 21783 df-cls 21784 df-nei 21861 df-lp 21899 df-perf 21900 df-cn 21990 df-cnp 21991 df-haus 22078 df-cmp 22150 df-tx 22325 df-hmeo 22518 df-fil 22609 df-fm 22701 df-flim 22702 df-flf 22703 df-xms 23085 df-ms 23086 df-tms 23087 df-cncf 23642 df-limc 24630 df-dv 24631 df-ulm 25136 df-log 25312 df-cxp 25313 df-lgam 25768 |
This theorem is referenced by: gam1 25814 |
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