wallispilem2 - Mathbox for Glauco Siliprandi

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Theorem wallispilem2 46037

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 12433	. . 3 ⊢ 0 ∈ ℕ₀
2		oveq2 7377	. . . . . . . 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 13345	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
5	4	sseli 3939	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
6	5	sincld 16074	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
8	7	exp0d 14081	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1)
9	3, 8	eqtrd 2764	. . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1)
10	9	itgeq2dv 25659	. . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥)
11		ioombl 25442	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
12		0re 11152	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		pire 26342	. . . . . . . 8 ⊢ π ∈ ℝ
14		ioovolcl 25447	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ)
15	12, 13, 14	mp2an 692	. . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ
16		ax-1cn 11102	. . . . . . 7 ⊢ 1 ∈ ℂ
17		itgconst 25696	. . . . . . 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 11164	. . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ
20	19	mullidi 11155	. . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π))
21		pipos 26344	. . . . . . . . . 10 ⊢ 0 < π
22	12, 13, 21	ltleii 11273	. . . . . . . . 9 ⊢ 0 ≤ π
23		volioo 25446	. . . . . . . . 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 11164	. . . . . . . . 9 ⊢ π ∈ ℂ
26	25	subid1i 11470	. . . . . . . 8 ⊢ (π − 0) = π
27	24, 26	eqtri 2752	. . . . . . 7 ⊢ (vol‘(0(,)π)) = π
28	20, 27	eqtri 2752	. . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π
29	18, 28	eqtri 2752	. . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π
30	10, 29	eqtrdi 2780	. . . 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 3467	. . . 4 ⊢ π ∈ V
33	30, 31, 32	fvmpt 6950	. . 3 ⊢ (0 ∈ ℕ₀ → (𝐼‘0) = π)
34	1, 33	ax-mp 5	. 2 ⊢ (𝐼‘0) = π
35		1nn0 12434	. . . 4 ⊢ 1 ∈ ℕ₀
36		simpl 482	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1)
37	36	oveq2d 7385	. . . . . . 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 14082	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥))
40	37, 39	eqtrd 2764	. . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥))
41	40	itgeq2dv 25659	. . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
42		itgex 25647	. . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V
43	41, 31, 42	fvmpt 6950	. . . 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 45923	. . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
46	44, 45	eqtri 2752	. 2 ⊢ (𝐼‘1) = 2
47		id 22	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘2))
48	31, 47	itgsinexp 45926	. 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 1540 ∈ wcel 2109 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 ℕ₀cn0 12418 ℤ_≥cuz 12769 (,)cioo 13282 ↑cexp 14002 sincsin 16005 πcpi 16008 volcvol 25340 ∫citg 25495

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-symdif 4212 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-ovol 25341 df-vol 25342 df-mbf 25496 df-itg1 25497 df-itg2 25498 df-ibl 25499 df-itg 25500 df-0p 25547 df-limc 25743 df-dv 25744

This theorem is referenced by: wallispilem3 46038 wallispilem4 46039

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