stirlingr - Mathbox for Glauco Siliprandi

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Theorem stirlingr 43521

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 43520 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 43520	. 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1
3		stirlingr.2	. . . 4 ⊢ 𝑅 = (⇝_𝑡‘(topGen‘ran (,)))
4		nnuz 12550	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5		1zzd 12281	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
6		eqid 2738	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))
7		nnnn0 12170	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
8		faccl 13925	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
9		nnre 11910	. . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ)
10	7, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ)
11		2re 11977	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
13		pire 25520	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ)
15	12, 14	remulcld 10936	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ)
16		nnre 11910	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
17	15, 16	remulcld 10936	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ)
18		0re 10908	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ)
20		2pos 12006	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2)
22	19, 12, 21	ltled 11053	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2)
23		pipos 25522	. . . . . . . . . . . . . . 15 ⊢ 0 < π
24	18, 13, 23	ltleii 11028	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π)
26	12, 14, 22, 25	mulge0d 11482	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π))
27	7	nn0ge0d 12226	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛)
28	15, 16, 26, 27	mulge0d 11482	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛))
29	17, 28	resqrtcld 15057	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ)
30		ere 15726	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ)
32		epos 15844	. . . . . . . . . . . . . 14 ⊢ 0 < e
33	18, 32	gtneii 11017	. . . . . . . . . . . . 13 ⊢ e ≠ 0
34	33	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0)
35	16, 31, 34	redivcld 11733	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ)
36	35, 7	reexpcld 13809	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ)
37	29, 36	remulcld 10936	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ)
38	1	fvmpt2 6868	. . . . . . . . 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 12664	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ⁺)
42		pirp 25523	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ⁺)
44	41, 43	rpmulcld 12717	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ⁺)
45		nnrp 12670	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
46	44, 45	rpmulcld 12717	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ⁺)
47	46	rpsqrtcld 15051	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ⁺)
48		epr 15845	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ⁺)
50	45, 49	rpdivcld 12718	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ⁺)
51		nnz 12272	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
52	50, 51	rpexpcld 13890	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ⁺)
53	47, 52	rpmulcld 12717	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ⁺)
54	39, 53	eqeltrd 2839	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ⁺)
55	10, 54	rerpdivcld 12732	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ)
56	6, 55	fmpti 6968	. . . . 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 43044	. . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1))
59	58	mptru 1546	. 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)
60	2, 59	mpbir 230	1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ↦ cmpt 5153 ran crn 5581 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 < clt 10940 ≤ cle 10941 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℝ⁺crp 12659 (,)cioo 13008 ↑cexp 13710 !cfa 13915 √csqrt 14872 ⇝ cli 15121 eceu 15700 πcpi 15704 topGenctg 17065 ⇝_𝑡clm 22285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-e 15706 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-lm 22288 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936 df-ulm 25441 df-log 25617 df-cxp 25618

This theorem is referenced by: (None)

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