![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 41930 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 41930 | . 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1 |
3 | stirlingr.2 | . . . 4 ⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,))) | |
4 | nnuz 12130 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 1zzd 11863 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
6 | eqid 2794 | . . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) | |
7 | nnnn0 11754 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
8 | faccl 13493 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (!‘𝑛) ∈ ℕ) | |
9 | nnre 11495 | . . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ) |
11 | 2re 11561 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ) |
13 | pire 24727 | . . . . . . . . . . . . . 14 ⊢ π ∈ ℝ | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ) |
15 | 12, 14 | remulcld 10520 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ) |
16 | nnre 11495 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
17 | 15, 16 | remulcld 10520 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ) |
18 | 0re 10492 | . . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ) |
20 | 2pos 11590 | . . . . . . . . . . . . . . 15 ⊢ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2) |
22 | 19, 12, 21 | ltled 10637 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2) |
23 | pipos 24729 | . . . . . . . . . . . . . . 15 ⊢ 0 < π | |
24 | 18, 13, 23 | ltleii 10612 | . . . . . . . . . . . . . 14 ⊢ 0 ≤ π |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π) |
26 | 12, 14, 22, 25 | mulge0d 11067 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π)) |
27 | 7 | nn0ge0d 11808 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛) |
28 | 15, 16, 26, 27 | mulge0d 11067 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛)) |
29 | 17, 28 | resqrtcld 14611 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ) |
30 | ere 15275 | . . . . . . . . . . . . 13 ⊢ e ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ) |
32 | epos 15393 | . . . . . . . . . . . . . 14 ⊢ 0 < e | |
33 | 18, 32 | gtneii 10601 | . . . . . . . . . . . . 13 ⊢ e ≠ 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0) |
35 | 16, 31, 34 | redivcld 11318 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ) |
36 | 35, 7 | reexpcld 13377 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ) |
37 | 29, 36 | remulcld 10520 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) |
38 | 1 | fvmpt2 6648 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
39 | 7, 37, 38 | syl2anc 584 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛))) |
40 | 2rp 12244 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ+) |
42 | pirp 24730 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ+) |
44 | 41, 43 | rpmulcld 12297 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ+) |
45 | nnrp 12250 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+) | |
46 | 44, 45 | rpmulcld 12297 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ+) |
47 | 46 | rpsqrtcld 14605 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ+) |
48 | epr 15394 | . . . . . . . . . . . 12 ⊢ e ∈ ℝ+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ+) |
50 | 45, 49 | rpdivcld 12298 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ+) |
51 | nnz 11854 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
52 | 50, 51 | rpexpcld 13458 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ+) |
53 | 47, 52 | rpmulcld 12297 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ+) |
54 | 39, 53 | eqeltrd 2882 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ+) |
55 | 10, 54 | rerpdivcld 12312 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ) |
56 | 6, 55 | fmpti 6742 | . . . . 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 41449 | . . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)) |
59 | 58 | mptru 1529 | . 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1) |
60 | 2, 59 | mpbir 232 | 1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1522 ⊤wtru 1523 ∈ wcel 2080 ≠ wne 2983 class class class wbr 4964 ↦ cmpt 5043 ran crn 5447 ⟶wf 6224 ‘cfv 6228 (class class class)co 7019 ℝcr 10385 0cc0 10386 1c1 10387 · cmul 10391 < clt 10524 ≤ cle 10525 / cdiv 11147 ℕcn 11488 2c2 11542 ℕ0cn0 11747 ℝ+crp 12239 (,)cioo 12588 ↑cexp 13279 !cfa 13483 √csqrt 14426 ⇝ cli 14675 eceu 15249 πcpi 15253 topGenctg 16540 ⇝𝑡clm 21518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cc 9706 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-symdif 4141 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-disj 4933 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-of 7270 df-ofr 7271 df-om 7440 df-1st 7548 df-2nd 7549 df-supp 7685 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-2o 7957 df-oadd 7960 df-omul 7961 df-er 8142 df-map 8261 df-pm 8262 df-ixp 8314 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fsupp 8683 df-fi 8724 df-sup 8755 df-inf 8756 df-oi 8823 df-dju 9179 df-card 9217 df-acn 9220 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-xnn0 11818 df-z 11832 df-dec 11949 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-e 15255 df-sin 15256 df-cos 15257 df-tan 15258 df-pi 15259 df-dvds 15441 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-lm 21521 df-haus 21607 df-cmp 21679 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-ovol 23748 df-vol 23749 df-mbf 23903 df-itg1 23904 df-itg2 23905 df-ibl 23906 df-itg 23907 df-0p 23954 df-limc 24147 df-dv 24148 df-ulm 24648 df-log 24821 df-cxp 24822 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |