stirlingr - Mathbox for Glauco Siliprandi

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Theorem stirlingr 46011

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 46010 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.)

**Hypotheses**
Ref	Expression
stirlingr.1	⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
stirlingr.2	⊢ 𝑅 = (⇝_𝑡‘(topGen‘ran (,)))

**Assertion**
Ref	Expression
stirlingr	⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1

**Proof of Theorem stirlingr**
Step	Hyp	Ref	Expression
1		stirlingr.1	. . 3 ⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
2	1	stirling 46010	. 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1
3		stirlingr.2	. . . 4 ⊢ 𝑅 = (⇝_𝑡‘(topGen‘ran (,)))
4		nnuz 12946	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5		1zzd 12674	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
6		eqid 2740	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))
7		nnnn0 12560	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
8		faccl 14332	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
9		nnre 12300	. . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ)
10	7, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ)
11		2re 12367	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
13		pire 26518	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ)
15	12, 14	remulcld 11320	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ)
16		nnre 12300	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
17	15, 16	remulcld 11320	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ)
18		0re 11292	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ)
20		2pos 12396	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2)
22	19, 12, 21	ltled 11438	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2)
23		pipos 26520	. . . . . . . . . . . . . . 15 ⊢ 0 < π
24	18, 13, 23	ltleii 11413	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π)
26	12, 14, 22, 25	mulge0d 11867	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π))
27	7	nn0ge0d 12616	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛)
28	15, 16, 26, 27	mulge0d 11867	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛))
29	17, 28	resqrtcld 15466	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ)
30		ere 16137	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ)
32		epos 16255	. . . . . . . . . . . . . 14 ⊢ 0 < e
33	18, 32	gtneii 11402	. . . . . . . . . . . . 13 ⊢ e ≠ 0
34	33	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0)
35	16, 31, 34	redivcld 12122	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ)
36	35, 7	reexpcld 14213	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ)
37	29, 36	remulcld 11320	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ)
38	1	fvmpt2 7040	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
39	7, 37, 38	syl2anc 583	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
40		2rp 13062	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ⁺)
42		pirp 26521	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ⁺)
44	41, 43	rpmulcld 13115	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ⁺)
45		nnrp 13068	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
46	44, 45	rpmulcld 13115	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ⁺)
47	46	rpsqrtcld 15460	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ⁺)
48		epr 16256	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ⁺)
50	45, 49	rpdivcld 13116	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ⁺)
51		nnz 12660	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
52	50, 51	rpexpcld 14296	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ⁺)
53	47, 52	rpmulcld 13115	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ⁺)
54	39, 53	eqeltrd 2844	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ⁺)
55	10, 54	rerpdivcld 13130	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ)
56	6, 55	fmpti 7146	. . . . 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 45534	. . 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 231	1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 ran crn 5701 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 < clt 11324 ≤ cle 11325 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℝ⁺crp 13057 (,)cioo 13407 ↑cexp 14112 !cfa 14322 √csqrt 15282 ⇝ cli 15530 eceu 16110 πcpi 16114 topGenctg 17497 ⇝_𝑡clm 23255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-dvds 16303 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-lm 23258 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-limc 25921 df-dv 25922 df-ulm 26438 df-log 26616 df-cxp 26617

This theorem is referenced by: (None)

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