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| Mirrors > Home > MPE Home > Th. List > logfac | Structured version Visualization version GIF version | ||
| Description: The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.) |
| Ref | Expression |
|---|---|
| logfac | ⊢ (𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0 12528 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 2 | rpmulcl 13058 | . . . . . 6 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑘 · 𝑛) ∈ ℝ+) | |
| 3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+)) → (𝑘 · 𝑛) ∈ ℝ+) |
| 4 | fvi 6985 | . . . . . . 7 ⊢ (𝑘 ∈ V → ( I ‘𝑘) = 𝑘) | |
| 5 | 4 | elv 3485 | . . . . . 6 ⊢ ( I ‘𝑘) = 𝑘 |
| 6 | elfznn 13593 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
| 7 | 6 | adantl 481 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ) |
| 8 | 7 | nnrpd 13075 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ+) |
| 9 | 5, 8 | eqeltrid 2845 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → ( I ‘𝑘) ∈ ℝ+) |
| 10 | elnnuz 12922 | . . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
| 11 | 10 | biimpi 216 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
| 12 | relogmul 26634 | . . . . . 6 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (log‘(𝑘 · 𝑛)) = ((log‘𝑘) + (log‘𝑛))) | |
| 13 | 12 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑘 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+)) → (log‘(𝑘 · 𝑛)) = ((log‘𝑘) + (log‘𝑛))) |
| 14 | 5 | fveq2i 6909 | . . . . . 6 ⊢ (log‘( I ‘𝑘)) = (log‘𝑘) |
| 15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘( I ‘𝑘)) = (log‘𝑘)) |
| 16 | 3, 9, 11, 13, 15 | seqhomo 14090 | . . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(seq1( · , I )‘𝑁)) = (seq1( + , log)‘𝑁)) |
| 17 | facnn 14314 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | |
| 18 | 17 | fveq2d 6910 | . . . 4 ⊢ (𝑁 ∈ ℕ → (log‘(!‘𝑁)) = (log‘(seq1( · , I )‘𝑁))) |
| 19 | eqidd 2738 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘𝑘) = (log‘𝑘)) | |
| 20 | relogcl 26617 | . . . . . . 7 ⊢ (𝑘 ∈ ℝ+ → (log‘𝑘) ∈ ℝ) | |
| 21 | 8, 20 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘𝑘) ∈ ℝ) |
| 22 | 21 | recnd 11289 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (1...𝑁)) → (log‘𝑘) ∈ ℂ) |
| 23 | 19, 11, 22 | fsumser 15766 | . . . 4 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (1...𝑁)(log‘𝑘) = (seq1( + , log)‘𝑁)) |
| 24 | 16, 18, 23 | 3eqtr4d 2787 | . . 3 ⊢ (𝑁 ∈ ℕ → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘)) |
| 25 | log1 26627 | . . . . 5 ⊢ (log‘1) = 0 | |
| 26 | sum0 15757 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ (log‘𝑘) = 0 | |
| 27 | 25, 26 | eqtr4i 2768 | . . . 4 ⊢ (log‘1) = Σ𝑘 ∈ ∅ (log‘𝑘) |
| 28 | fveq2 6906 | . . . . . 6 ⊢ (𝑁 = 0 → (!‘𝑁) = (!‘0)) | |
| 29 | fac0 14315 | . . . . . 6 ⊢ (!‘0) = 1 | |
| 30 | 28, 29 | eqtrdi 2793 | . . . . 5 ⊢ (𝑁 = 0 → (!‘𝑁) = 1) |
| 31 | 30 | fveq2d 6910 | . . . 4 ⊢ (𝑁 = 0 → (log‘(!‘𝑁)) = (log‘1)) |
| 32 | oveq2 7439 | . . . . . 6 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0)) | |
| 33 | fz10 13585 | . . . . . 6 ⊢ (1...0) = ∅ | |
| 34 | 32, 33 | eqtrdi 2793 | . . . . 5 ⊢ (𝑁 = 0 → (1...𝑁) = ∅) |
| 35 | 34 | sumeq1d 15736 | . . . 4 ⊢ (𝑁 = 0 → Σ𝑘 ∈ (1...𝑁)(log‘𝑘) = Σ𝑘 ∈ ∅ (log‘𝑘)) |
| 36 | 27, 31, 35 | 3eqtr4a 2803 | . . 3 ⊢ (𝑁 = 0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘)) |
| 37 | 24, 36 | jaoi 858 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘)) |
| 38 | 1, 37 | sylbi 217 | 1 ⊢ (𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 I cid 5577 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 ℤ≥cuz 12878 ℝ+crp 13034 ...cfz 13547 seqcseq 14042 !cfa 14312 Σcsu 15722 logclog 26596 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 |
| This theorem is referenced by: birthdaylem2 26995 logfac2 27261 logfaclbnd 27266 logfacbnd3 27267 |
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