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 42381 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 42381 | . 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1 |
3 | stirlingr.2 | . . . 4 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
4 | nnuz 12284 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 1zzd 12016 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
6 | eqid 2823 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) | |
7 | nnnn0 11907 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
8 | faccl 13646 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ) | |
9 | nnre 11647 | . . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ) |
11 | 2re 11714 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ) |
13 | pire 25046 | . . . . . . . . . . . . . 14 ⊢ π ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ) |
15 | 12, 14 | remulcld 10673 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ) |
16 | nnre 11647 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
17 | 15, 16 | remulcld 10673 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ) |
18 | 0re 10645 | . . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ) |
20 | 2pos 11743 | . . . . . . . . . . . . . . 15 ⊢ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2) |
22 | 19, 12, 21 | ltled 10790 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2) |
23 | pipos 25048 | . . . . . . . . . . . . . . 15 ⊢ 0 < π | |
24 | 18, 13, 23 | ltleii 10765 | . . . . . . . . . . . . . 14 ⊢ 0 ≤ π |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π) |
26 | 12, 14, 22, 25 | mulge0d 11219 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π)) |
27 | 7 | nn0ge0d 11961 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛) |
28 | 15, 16, 26, 27 | mulge0d 11219 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛)) |
29 | 17, 28 | resqrtcld 14779 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ) |
30 | ere 15444 | . . . . . . . . . . . . 13 ⊢ e ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ) |
32 | epos 15562 | . . . . . . . . . . . . . 14 ⊢ 0 < e | |
33 | 18, 32 | gtneii 10754 | . . . . . . . . . . . . 13 ⊢ e ≠ 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0) |
35 | 16, 31, 34 | redivcld 11470 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ) |
36 | 35, 7 | reexpcld 13530 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ) |
37 | 29, 36 | remulcld 10673 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) |
38 | 1 | fvmpt2 6781 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
39 | 7, 37, 38 | syl2anc 586 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
40 | 2rp 12397 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ+) |
42 | pirp 25049 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ+) |
44 | 41, 43 | rpmulcld 12450 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ+) |
45 | nnrp 12403 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
46 | 44, 45 | rpmulcld 12450 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ+) |
47 | 46 | rpsqrtcld 14773 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ+) |
48 | epr 15563 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ+) |
50 | 45, 49 | rpdivcld 12451 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ+) |
51 | nnz 12007 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
52 | 50, 51 | rpexpcld 13611 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ+) |
53 | 47, 52 | rpmulcld 12450 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ+) |
54 | 39, 53 | eqeltrd 2915 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ+) |
55 | 10, 54 | rerpdivcld 12465 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ) |
56 | 6, 55 | fmpti 6878 | . . . . 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 41901 | . . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)) |
59 | 58 | mptru 1544 | . 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1) |
60 | 2, 59 | mpbir 233 | 1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 · cmul 10544 < clt 10677 ≤ cle 10678 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ0cn0 11900 ℝ+crp 12392 (,)cioo 12741 ↑cexp 13432 !cfa 13636 √csqrt 14594 ⇝ cli 14843 eceu 15418 πcpi 15422 topGenctg 16713 ⇝𝑡clm 21836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-symdif 4221 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-e 15424 df-sin 15425 df-cos 15426 df-tan 15427 df-pi 15428 df-dvds 15610 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-lm 21839 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-ovol 24067 df-vol 24068 df-mbf 24222 df-itg1 24223 df-itg2 24224 df-ibl 24225 df-itg 24226 df-0p 24273 df-limc 24466 df-dv 24467 df-ulm 24967 df-log 25142 df-cxp 25143 |
This theorem is referenced by: (None) |
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