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 43212 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 43212 | . 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1 |
3 | stirlingr.2 | . . . 4 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
4 | nnuz 12375 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 1zzd 12106 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
6 | eqid 2739 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) | |
7 | nnnn0 11995 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
8 | faccl 13747 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ) | |
9 | nnre 11735 | . . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ) |
11 | 2re 11802 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ) |
13 | pire 25215 | . . . . . . . . . . . . . 14 ⊢ π ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ) |
15 | 12, 14 | remulcld 10761 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ) |
16 | nnre 11735 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
17 | 15, 16 | remulcld 10761 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ) |
18 | 0re 10733 | . . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ) |
20 | 2pos 11831 | . . . . . . . . . . . . . . 15 ⊢ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2) |
22 | 19, 12, 21 | ltled 10878 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2) |
23 | pipos 25217 | . . . . . . . . . . . . . . 15 ⊢ 0 < π | |
24 | 18, 13, 23 | ltleii 10853 | . . . . . . . . . . . . . 14 ⊢ 0 ≤ π |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π) |
26 | 12, 14, 22, 25 | mulge0d 11307 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π)) |
27 | 7 | nn0ge0d 12051 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛) |
28 | 15, 16, 26, 27 | mulge0d 11307 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛)) |
29 | 17, 28 | resqrtcld 14879 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ) |
30 | ere 15546 | . . . . . . . . . . . . 13 ⊢ e ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ) |
32 | epos 15664 | . . . . . . . . . . . . . 14 ⊢ 0 < e | |
33 | 18, 32 | gtneii 10842 | . . . . . . . . . . . . 13 ⊢ e ≠ 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0) |
35 | 16, 31, 34 | redivcld 11558 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ) |
36 | 35, 7 | reexpcld 13631 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ) |
37 | 29, 36 | remulcld 10761 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) |
38 | 1 | fvmpt2 6798 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
39 | 7, 37, 38 | syl2anc 587 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
40 | 2rp 12489 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ+) |
42 | pirp 25218 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ+) |
44 | 41, 43 | rpmulcld 12542 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ+) |
45 | nnrp 12495 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
46 | 44, 45 | rpmulcld 12542 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ+) |
47 | 46 | rpsqrtcld 14873 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ+) |
48 | epr 15665 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ+) |
50 | 45, 49 | rpdivcld 12543 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ+) |
51 | nnz 12097 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
52 | 50, 51 | rpexpcld 13712 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ+) |
53 | 47, 52 | rpmulcld 12542 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ+) |
54 | 39, 53 | eqeltrd 2834 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ+) |
55 | 10, 54 | rerpdivcld 12557 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ) |
56 | 6, 55 | fmpti 6898 | . . . . 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 42736 | . . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)) |
59 | 58 | mptru 1549 | . 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1) |
60 | 2, 59 | mpbir 234 | 1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ↦ cmpt 5120 ran crn 5536 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 0cc0 10627 1c1 10628 · cmul 10632 < clt 10765 ≤ cle 10766 / cdiv 11387 ℕcn 11728 2c2 11783 ℕ0cn0 11988 ℝ+crp 12484 (,)cioo 12833 ↑cexp 13533 !cfa 13737 √csqrt 14694 ⇝ cli 14943 eceu 15520 πcpi 15524 topGenctg 16826 ⇝𝑡clm 21989 |
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-cc 9947 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-symdif 4143 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-disj 5006 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-ofr 7438 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-omul 8148 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-acn 9456 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-xnn0 12061 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-e 15526 df-sin 15527 df-cos 15528 df-tan 15529 df-pi 15530 df-dvds 15712 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-lm 21992 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-ovol 24228 df-vol 24229 df-mbf 24383 df-itg1 24384 df-itg2 24385 df-ibl 24386 df-itg 24387 df-0p 24434 df-limc 24630 df-dv 24631 df-ulm 25136 df-log 25312 df-cxp 25313 |
This theorem is referenced by: (None) |
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