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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stirlingr | Structured version Visualization version GIF version |
Description: Stirling's approximation formula for 𝑛 factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 45615 is proven for convergence in the topology of complex numbers. The variable 𝑅 is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
stirlingr.1 | ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
stirlingr.2 | ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) |
Ref | Expression |
---|---|
stirlingr | ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stirlingr.1 | . . 3 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) | |
2 | 1 | stirling 45615 | . 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1 |
3 | stirlingr.2 | . . . 4 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
4 | nnuz 12898 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 1zzd 12626 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
6 | eqid 2725 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) | |
7 | nnnn0 12512 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
8 | faccl 14278 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ) | |
9 | nnre 12252 | . . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ) |
11 | 2re 12319 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ) |
13 | pire 26438 | . . . . . . . . . . . . . 14 ⊢ π ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ) |
15 | 12, 14 | remulcld 11276 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ) |
16 | nnre 12252 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
17 | 15, 16 | remulcld 11276 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ) |
18 | 0re 11248 | . . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ) |
20 | 2pos 12348 | . . . . . . . . . . . . . . 15 ⊢ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2) |
22 | 19, 12, 21 | ltled 11394 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2) |
23 | pipos 26440 | . . . . . . . . . . . . . . 15 ⊢ 0 < π | |
24 | 18, 13, 23 | ltleii 11369 | . . . . . . . . . . . . . 14 ⊢ 0 ≤ π |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π) |
26 | 12, 14, 22, 25 | mulge0d 11823 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π)) |
27 | 7 | nn0ge0d 12568 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛) |
28 | 15, 16, 26, 27 | mulge0d 11823 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛)) |
29 | 17, 28 | resqrtcld 15400 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ) |
30 | ere 16069 | . . . . . . . . . . . . 13 ⊢ e ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ) |
32 | epos 16187 | . . . . . . . . . . . . . 14 ⊢ 0 < e | |
33 | 18, 32 | gtneii 11358 | . . . . . . . . . . . . 13 ⊢ e ≠ 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0) |
35 | 16, 31, 34 | redivcld 12075 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ) |
36 | 35, 7 | reexpcld 14163 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ) |
37 | 29, 36 | remulcld 11276 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) |
38 | 1 | fvmpt2 7015 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
39 | 7, 37, 38 | syl2anc 582 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
40 | 2rp 13014 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ+) |
42 | pirp 26441 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ+) |
44 | 41, 43 | rpmulcld 13067 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ+) |
45 | nnrp 13020 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
46 | 44, 45 | rpmulcld 13067 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ+) |
47 | 46 | rpsqrtcld 15394 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ+) |
48 | epr 16188 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ+) |
50 | 45, 49 | rpdivcld 13068 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ+) |
51 | nnz 12612 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
52 | 50, 51 | rpexpcld 14245 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ+) |
53 | 47, 52 | rpmulcld 13067 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ+) |
54 | 39, 53 | eqeltrd 2825 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ+) |
55 | 10, 54 | rerpdivcld 13082 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ) |
56 | 6, 55 | fmpti 7121 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))):ℕ⟶ℝ |
57 | 56 | a1i 11 | . . . 4 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))):ℕ⟶ℝ) |
58 | 3, 4, 5, 57 | climreeq 45139 | . . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)) |
59 | 58 | mptru 1540 | . 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1) |
60 | 2, 59 | mpbir 230 | 1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ↦ cmpt 5232 ran crn 5679 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 0cc0 11140 1c1 11141 · cmul 11145 < clt 11280 ≤ cle 11281 / cdiv 11903 ℕcn 12245 2c2 12300 ℕ0cn0 12505 ℝ+crp 13009 (,)cioo 13359 ↑cexp 14062 !cfa 14268 √csqrt 15216 ⇝ cli 15464 eceu 16042 πcpi 16046 topGenctg 17422 ⇝𝑡clm 23174 |
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-cc 10460 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-symdif 4241 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-disj 5115 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-ofr 7686 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-omul 8492 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-acn 9967 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-xnn0 12578 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-e 16048 df-sin 16049 df-cos 16050 df-tan 16051 df-pi 16052 df-dvds 16235 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-lm 23177 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-ovol 25437 df-vol 25438 df-mbf 25592 df-itg1 25593 df-itg2 25594 df-ibl 25595 df-itg 25596 df-0p 25643 df-limc 25839 df-dv 25840 df-ulm 26358 df-log 26535 df-cxp 26536 |
This theorem is referenced by: (None) |
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