wallispilem2 - Mathbox for Glauco Siliprandi

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Theorem wallispilem2 46174

Description: A first set of properties for the sequence 𝐼 that will be used in the proof of the Wallis product formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypothesis**
Ref	Expression
wallispilem2.1	⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)

**Assertion**
Ref	Expression
wallispilem2	⊢ ((𝐼‘0) = π ∧ (𝐼‘1) = 2 ∧ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))))

Distinct variable group: 𝑥,𝑛,𝑁 Allowed substitution hints: 𝐼(𝑥,𝑛)

**Proof of Theorem wallispilem2**
Step	Hyp	Ref	Expression
1		0nn0 12396	. . 3 ⊢ 0 ∈ ℕ₀
2		oveq2 7354	. . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
3	2	adantr 480	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
4		ioosscn 13308	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
5	4	sseli 3925	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
6	5	sincld 16039	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
8	7	exp0d 14047	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1)
9	3, 8	eqtrd 2766	. . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1)
10	9	itgeq2dv 25710	. . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥)
11		ioombl 25493	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
12		0re 11114	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		pire 26393	. . . . . . . 8 ⊢ π ∈ ℝ
14		ioovolcl 25498	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ)
15	12, 13, 14	mp2an 692	. . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ
16		ax-1cn 11064	. . . . . . 7 ⊢ 1 ∈ ℂ
17		itgconst 25747	. . . . . . 7 ⊢ (((0(,)π) ∈ dom vol ∧ (vol‘(0(,)π)) ∈ ℝ ∧ 1 ∈ ℂ) → ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π))))
18	11, 15, 16, 17	mp3an 1463	. . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π)))
19	15	recni 11126	. . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ
20	19	mullidi 11117	. . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π))
21		pipos 26395	. . . . . . . . . 10 ⊢ 0 < π
22	12, 13, 21	ltleii 11236	. . . . . . . . 9 ⊢ 0 ≤ π
23		volioo 25497	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0))
24	12, 13, 22, 23	mp3an 1463	. . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0)
25	13	recni 11126	. . . . . . . . 9 ⊢ π ∈ ℂ
26	25	subid1i 11433	. . . . . . . 8 ⊢ (π − 0) = π
27	24, 26	eqtri 2754	. . . . . . 7 ⊢ (vol‘(0(,)π)) = π
28	20, 27	eqtri 2754	. . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π
29	18, 28	eqtri 2754	. . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π
30	10, 29	eqtrdi 2782	. . . 4 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = π)
31		wallispilem2.1	. . . 4 ⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)
32	13	elexi 3459	. . . 4 ⊢ π ∈ V
33	30, 31, 32	fvmpt 6929	. . 3 ⊢ (0 ∈ ℕ₀ → (𝐼‘0) = π)
34	1, 33	ax-mp 5	. 2 ⊢ (𝐼‘0) = π
35		1nn0 12397	. . . 4 ⊢ 1 ∈ ℕ₀
36		simpl 482	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1)
37	36	oveq2d 7362	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1))
38	6	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
39	38	exp1d 14048	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥))
40	37, 39	eqtrd 2766	. . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥))
41	40	itgeq2dv 25710	. . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
42		itgex 25698	. . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V
43	41, 31, 42	fvmpt 6929	. . . 4 ⊢ (1 ∈ ℕ₀ → (𝐼‘1) = ∫(0(,)π)(sin‘𝑥) d𝑥)
44	35, 43	ax-mp 5	. . 3 ⊢ (𝐼‘1) = ∫(0(,)π)(sin‘𝑥) d𝑥
45		itgsin0pi 46060	. . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
46	44, 45	eqtri 2754	. 2 ⊢ (𝐼‘1) = 2
47		id 22	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘2))
48	31, 47	itgsinexp 46063	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
49	34, 46, 48	3pm3.2i 1340	1 ⊢ ((𝐼‘0) = π ∧ (𝐼‘1) = 2 ∧ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 · cmul 11011 ≤ cle 11147 − cmin 11344 / cdiv 11774 2c2 12180 ℕ₀cn0 12381 ℤ_≥cuz 12732 (,)cioo 13245 ↑cexp 13968 sincsin 15970 πcpi 15973 volcvol 25391 ∫citg 25546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-symdif 4200 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 df-itg2 25549 df-ibl 25550 df-itg 25551 df-0p 25598 df-limc 25794 df-dv 25795

This theorem is referenced by: wallispilem3 46175 wallispilem4 46176

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