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 42394 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 42394 | . 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1 |
3 | stirlingr.2 | . . . 4 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
4 | nnuz 12282 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 1zzd 12014 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
6 | eqid 2821 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) | |
7 | nnnn0 11905 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
8 | faccl 13644 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ) | |
9 | nnre 11645 | . . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ) |
11 | 2re 11712 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ) |
13 | pire 25044 | . . . . . . . . . . . . . 14 ⊢ π ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ) |
15 | 12, 14 | remulcld 10671 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ) |
16 | nnre 11645 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
17 | 15, 16 | remulcld 10671 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ) |
18 | 0re 10643 | . . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ) |
20 | 2pos 11741 | . . . . . . . . . . . . . . 15 ⊢ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2) |
22 | 19, 12, 21 | ltled 10788 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2) |
23 | pipos 25046 | . . . . . . . . . . . . . . 15 ⊢ 0 < π | |
24 | 18, 13, 23 | ltleii 10763 | . . . . . . . . . . . . . 14 ⊢ 0 ≤ π |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π) |
26 | 12, 14, 22, 25 | mulge0d 11217 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π)) |
27 | 7 | nn0ge0d 11959 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛) |
28 | 15, 16, 26, 27 | mulge0d 11217 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛)) |
29 | 17, 28 | resqrtcld 14777 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ) |
30 | ere 15442 | . . . . . . . . . . . . 13 ⊢ e ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ) |
32 | epos 15560 | . . . . . . . . . . . . . 14 ⊢ 0 < e | |
33 | 18, 32 | gtneii 10752 | . . . . . . . . . . . . 13 ⊢ e ≠ 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0) |
35 | 16, 31, 34 | redivcld 11468 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ) |
36 | 35, 7 | reexpcld 13528 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ) |
37 | 29, 36 | remulcld 10671 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) |
38 | 1 | fvmpt2 6779 | . . . . . . . . 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 12395 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ+) |
42 | pirp 25047 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ+) |
44 | 41, 43 | rpmulcld 12448 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ+) |
45 | nnrp 12401 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
46 | 44, 45 | rpmulcld 12448 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ+) |
47 | 46 | rpsqrtcld 14771 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ+) |
48 | epr 15561 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ+) |
50 | 45, 49 | rpdivcld 12449 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ+) |
51 | nnz 12005 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
52 | 50, 51 | rpexpcld 13609 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ+) |
53 | 47, 52 | rpmulcld 12448 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ+) |
54 | 39, 53 | eqeltrd 2913 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ+) |
55 | 10, 54 | rerpdivcld 12463 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ) |
56 | 6, 55 | fmpti 6876 | . . . . 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 41914 | . . 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 3016 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 ≤ cle 10676 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ0cn0 11898 ℝ+crp 12390 (,)cioo 12739 ↑cexp 13430 !cfa 13634 √csqrt 14592 ⇝ cli 14841 eceu 15416 πcpi 15420 topGenctg 16711 ⇝𝑡clm 21834 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-symdif 4219 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-lm 21837 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 df-itg2 24222 df-ibl 24223 df-itg 24224 df-0p 24271 df-limc 24464 df-dv 24465 df-ulm 24965 df-log 25140 df-cxp 25141 |
This theorem is referenced by: (None) |
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