wallispilem2 - Mathbox for Glauco Siliprandi

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Theorem wallispilem2 43836

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 12298	. . 3 ⊢ 0 ∈ ℕ₀
2		oveq2 7315	. . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
3	2	adantr 482	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
4		ioosscn 13191	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
5	4	sseli 3922	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
6	5	sincld 15888	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
7	6	adantl 483	. . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
8	7	exp0d 13908	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1)
9	3, 8	eqtrd 2776	. . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1)
10	9	itgeq2dv 24995	. . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥)
11		ioombl 24778	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
12		0re 11027	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		pire 25664	. . . . . . . 8 ⊢ π ∈ ℝ
14		ioovolcl 24783	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ)
15	12, 13, 14	mp2an 690	. . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ
16		ax-1cn 10979	. . . . . . 7 ⊢ 1 ∈ ℂ
17		itgconst 25032	. . . . . . 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 1461	. . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π)))
19	15	recni 11039	. . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ
20	19	mulid2i 11030	. . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π))
21		pipos 25666	. . . . . . . . . 10 ⊢ 0 < π
22	12, 13, 21	ltleii 11148	. . . . . . . . 9 ⊢ 0 ≤ π
23		volioo 24782	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0))
24	12, 13, 22, 23	mp3an 1461	. . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0)
25	13	recni 11039	. . . . . . . . 9 ⊢ π ∈ ℂ
26	25	subid1i 11343	. . . . . . . 8 ⊢ (π − 0) = π
27	24, 26	eqtri 2764	. . . . . . 7 ⊢ (vol‘(0(,)π)) = π
28	20, 27	eqtri 2764	. . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π
29	18, 28	eqtri 2764	. . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π
30	10, 29	eqtrdi 2792	. . . 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 3456	. . . 4 ⊢ π ∈ V
33	30, 31, 32	fvmpt 6907	. . 3 ⊢ (0 ∈ ℕ₀ → (𝐼‘0) = π)
34	1, 33	ax-mp 5	. 2 ⊢ (𝐼‘0) = π
35		1nn0 12299	. . . 4 ⊢ 1 ∈ ℕ₀
36		simpl 484	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1)
37	36	oveq2d 7323	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1))
38	6	adantl 483	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
39	38	exp1d 13909	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥))
40	37, 39	eqtrd 2776	. . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥))
41	40	itgeq2dv 24995	. . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
42		itgex 24984	. . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V
43	41, 31, 42	fvmpt 6907	. . . 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 43722	. . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
46	44, 45	eqtri 2764	. 2 ⊢ (𝐼‘1) = 2
47		id 22	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘2))
48	31, 47	itgsinexp 43725	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
49	34, 46, 48	3pm3.2i 1339	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 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 ↦ cmpt 5164 dom cdm 5600 ‘cfv 6458 (class class class)co 7307 ℂcc 10919 ℝcr 10920 0cc0 10921 1c1 10922 · cmul 10926 ≤ cle 11060 − cmin 11255 / cdiv 11682 2c2 12078 ℕ₀cn0 12283 ℤ_≥cuz 12632 (,)cioo 13129 ↑cexp 13832 sincsin 15822 πcpi 15825 volcvol 24676 ∫citg 24831

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9447 ax-cc 10241 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 ax-addf 11000 ax-mulf 11001

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-symdif 4182 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-disj 5047 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-ofr 7566 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-oadd 8332 df-omul 8333 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9177 df-fi 9218 df-sup 9249 df-inf 9250 df-oi 9317 df-dju 9707 df-card 9745 df-acn 9748 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-5 12089 df-6 12090 df-7 12091 df-8 12092 df-9 12093 df-n0 12284 df-z 12370 df-dec 12488 df-uz 12633 df-q 12739 df-rp 12781 df-xneg 12898 df-xadd 12899 df-xmul 12900 df-ioo 13133 df-ioc 13134 df-ico 13135 df-icc 13136 df-fz 13290 df-fzo 13433 df-fl 13562 df-mod 13640 df-seq 13772 df-exp 13833 df-fac 14038 df-bc 14067 df-hash 14095 df-shft 14827 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-limsup 15229 df-clim 15246 df-rlim 15247 df-sum 15447 df-ef 15826 df-sin 15828 df-cos 15829 df-pi 15831 df-struct 16897 df-sets 16914 df-slot 16932 df-ndx 16944 df-base 16962 df-ress 16991 df-plusg 17024 df-mulr 17025 df-starv 17026 df-sca 17027 df-vsca 17028 df-ip 17029 df-tset 17030 df-ple 17031 df-ds 17033 df-unif 17034 df-hom 17035 df-cco 17036 df-rest 17182 df-topn 17183 df-0g 17201 df-gsum 17202 df-topgen 17203 df-pt 17204 df-prds 17207 df-xrs 17262 df-qtop 17267 df-imas 17268 df-xps 17270 df-mre 17344 df-mrc 17345 df-acs 17347 df-mgm 18375 df-sgrp 18424 df-mnd 18435 df-submnd 18480 df-mulg 18750 df-cntz 18972 df-cmn 19437 df-psmet 20638 df-xmet 20639 df-met 20640 df-bl 20641 df-mopn 20642 df-fbas 20643 df-fg 20644 df-cnfld 20647 df-top 22092 df-topon 22109 df-topsp 22131 df-bases 22145 df-cld 22219 df-ntr 22220 df-cls 22221 df-nei 22298 df-lp 22336 df-perf 22337 df-cn 22427 df-cnp 22428 df-haus 22515 df-cmp 22587 df-tx 22762 df-hmeo 22955 df-fil 23046 df-fm 23138 df-flim 23139 df-flf 23140 df-xms 23522 df-ms 23523 df-tms 23524 df-cncf 24090 df-ovol 24677 df-vol 24678 df-mbf 24832 df-itg1 24833 df-itg2 24834 df-ibl 24835 df-itg 24836 df-0p 24883 df-limc 25079 df-dv 25080

This theorem is referenced by: wallispilem3 43837 wallispilem4 43838

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