stirlingr - Mathbox for Glauco Siliprandi

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Theorem stirlingr 46046

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 46045 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 46045	. 2 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1
3		stirlingr.2	. . . 4 ⊢ 𝑅 = (⇝_𝑡‘(topGen‘ran (,)))
4		nnuz 12919	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5		1zzd 12646	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
6		eqid 2735	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛)))
7		nnnn0 12531	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
8		faccl 14319	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → (!‘𝑛) ∈ ℕ)
9		nnre 12271	. . . . . . . 8 ⊢ ((!‘𝑛) ∈ ℕ → (!‘𝑛) ∈ ℝ)
10	7, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (!‘𝑛) ∈ ℝ)
11		2re 12338	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ)
13		pire 26515	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ)
15	12, 14	remulcld 11289	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ)
16		nnre 12271	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
17	15, 16	remulcld 11289	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ)
18		0re 11261	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ∈ ℝ)
20		2pos 12367	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 < 2)
22	19, 12, 21	ltled 11407	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ 2)
23		pipos 26517	. . . . . . . . . . . . . . 15 ⊢ 0 < π
24	18, 13, 23	ltleii 11382	. . . . . . . . . . . . . 14 ⊢ 0 ≤ π
25	24	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ π)
26	12, 14, 22, 25	mulge0d 11838	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (2 · π))
27	7	nn0ge0d 12588	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ 𝑛)
28	15, 16, 26, 27	mulge0d 11838	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 0 ≤ ((2 · π) · 𝑛))
29	17, 28	resqrtcld 15453	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ)
30		ere 16122	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ)
32		epos 16240	. . . . . . . . . . . . . 14 ⊢ 0 < e
33	18, 32	gtneii 11371	. . . . . . . . . . . . 13 ⊢ e ≠ 0
34	33	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → e ≠ 0)
35	16, 31, 34	redivcld 12093	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ)
36	35, 7	reexpcld 14200	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ)
37	29, 36	remulcld 11289	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)) ∈ ℝ)
38	1	fvmpt2 7027	. . . . . . . . 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 13037	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℝ⁺)
42		pirp 26518	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ⁺
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → π ∈ ℝ⁺)
44	41, 43	rpmulcld 13091	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · π) ∈ ℝ⁺)
45		nnrp 13044	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
46	44, 45	rpmulcld 13091	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((2 · π) · 𝑛) ∈ ℝ⁺)
47	46	rpsqrtcld 15447	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (√‘((2 · π) · 𝑛)) ∈ ℝ⁺)
48		epr 16241	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → e ∈ ℝ⁺)
50	45, 49	rpdivcld 13092	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (𝑛 / e) ∈ ℝ⁺)
51		nnz 12632	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
52	50, 51	rpexpcld 14283	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑛 / e)↑𝑛) ∈ ℝ⁺)
53	47, 52	rpmulcld 13091	. . . . . . . 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 13106	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((!‘𝑛) / (𝑆‘𝑛)) ∈ ℝ)
56	6, 55	fmpti 7132	. . . . 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 45569	. . 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 2106 ≠ wne 2938 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℝ⁺crp 13032 (,)cioo 13384 ↑cexp 14099 !cfa 14309 √csqrt 15269 ⇝ cli 15517 eceu 16095 πcpi 16099 topGenctg 17484 ⇝_𝑡clm 23250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-lm 23253 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-limc 25916 df-dv 25917 df-ulm 26435 df-log 26613 df-cxp 26614

This theorem is referenced by: (None)

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