stirlingr - Mathbox for Glauco Siliprandi

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Theorem stirlingr 43213

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 43212 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 43212	. 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1
3		stirlingr.2	. . . 4 ⊢ 𝑅 = (⇝_𝑡‘(topGen‘ran (,)))
4		nnuz 12375	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5		1zzd 12106	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
6		eqid 2739	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))
7		nnnn0 11995	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
8		faccl 13747	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
9		nnre 11735	. . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ)
10	7, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ)
11		2re 11802	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
13		pire 25215	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ)
15	12, 14	remulcld 10761	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ)
16		nnre 11735	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
17	15, 16	remulcld 10761	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ)
18		0re 10733	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ)
20		2pos 11831	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2)
22	19, 12, 21	ltled 10878	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2)
23		pipos 25217	. . . . . . . . . . . . . . 15 ⊢ 0 < π
24	18, 13, 23	ltleii 10853	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π)
26	12, 14, 22, 25	mulge0d 11307	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π))
27	7	nn0ge0d 12051	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛)
28	15, 16, 26, 27	mulge0d 11307	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛))
29	17, 28	resqrtcld 14879	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ)
30		ere 15546	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ)
32		epos 15664	. . . . . . . . . . . . . 14 ⊢ 0 < e
33	18, 32	gtneii 10842	. . . . . . . . . . . . 13 ⊢ e ≠ 0
34	33	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0)
35	16, 31, 34	redivcld 11558	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ)
36	35, 7	reexpcld 13631	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ)
37	29, 36	remulcld 10761	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ)
38	1	fvmpt2 6798	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ) → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
39	7, 37, 38	syl2anc 587	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) = ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))
40		2rp 12489	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ⁺)
42		pirp 25218	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ⁺)
44	41, 43	rpmulcld 12542	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ⁺)
45		nnrp 12495	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
46	44, 45	rpmulcld 12542	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ⁺)
47	46	rpsqrtcld 14873	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ⁺)
48		epr 15665	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ⁺)
50	45, 49	rpdivcld 12543	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ⁺)
51		nnz 12097	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
52	50, 51	rpexpcld 13712	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ⁺)
53	47, 52	rpmulcld 12542	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ⁺)
54	39, 53	eqeltrd 2834	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑆‘𝑛) ∈ ℝ⁺)
55	10, 54	rerpdivcld 12557	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ)
56	6, 55	fmpti 6898	. . . . 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 42736	. . 3 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1))
59	58	mptru 1549	. 2 ⊢ ((𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1 ↔ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1)
60	2, 59	mpbir 234	1 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))𝑅1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ↦ cmpt 5120 ran crn 5536 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 0cc0 10627 1c1 10628 · cmul 10632 < clt 10765 ≤ cle 10766 / cdiv 11387 ℕcn 11728 2c2 11783 ℕ₀cn0 11988 ℝ⁺crp 12484 (,)cioo 12833 ↑cexp 13533 !cfa 13737 √csqrt 14694 ⇝ cli 14943 eceu 15520 πcpi 15524 topGenctg 16826 ⇝_𝑡clm 21989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cc 9947 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-symdif 4143 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-disj 5006 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-ofr 7438 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-oadd 8147 df-omul 8148 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-dju 9415 df-card 9453 df-acn 9456 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-xnn0 12061 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ioc 12838 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-fac 13738 df-bc 13767 df-hash 13795 df-shft 14528 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-limsup 14930 df-clim 14947 df-rlim 14948 df-sum 15148 df-ef 15525 df-e 15526 df-sin 15527 df-cos 15528 df-tan 15529 df-pi 15530 df-dvds 15712 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-hom 16704 df-cco 16705 df-rest 16811 df-topn 16812 df-0g 16830 df-gsum 16831 df-topgen 16832 df-pt 16833 df-prds 16836 df-xrs 16890 df-qtop 16895 df-imas 16896 df-xps 16898 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-mulg 18355 df-cntz 18577 df-cmn 19038 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-fbas 20226 df-fg 20227 df-cnfld 20230 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-cld 21782 df-ntr 21783 df-cls 21784 df-nei 21861 df-lp 21899 df-perf 21900 df-cn 21990 df-cnp 21991 df-lm 21992 df-haus 22078 df-cmp 22150 df-tx 22325 df-hmeo 22518 df-fil 22609 df-fm 22701 df-flim 22702 df-flf 22703 df-xms 23085 df-ms 23086 df-tms 23087 df-cncf 23642 df-ovol 24228 df-vol 24229 df-mbf 24383 df-itg1 24384 df-itg2 24385 df-ibl 24386 df-itg 24387 df-0p 24434 df-limc 24630 df-dv 24631 df-ulm 25136 df-log 25312 df-cxp 25313

This theorem is referenced by: (None)

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