wallispilem2 - Mathbox for Glauco Siliprandi

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Theorem wallispilem2 46509

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 12443	. . 3 ⊢ 0 ∈ ℕ₀
2		oveq2 7364	. . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
4		ioosscn 13352	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
5	4	sseli 3911	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
6	5	sincld 16088	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
7	6	adantl 482	. . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
8	7	exp0d 14093	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1)
9	3, 8	eqtrd 2774	. . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1)
10	9	itgeq2dv 25767	. . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥)
11		ioombl 25550	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
12		0re 11137	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		pire 26439	. . . . . . . 8 ⊢ π ∈ ℝ
14		ioovolcl 25555	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ)
15	12, 13, 14	mp2an 698	. . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ
16		ax-1cn 11087	. . . . . . 7 ⊢ 1 ∈ ℂ
17		itgconst 25804	. . . . . . 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 1469	. . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π)))
19	15	recni 11150	. . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ
20	19	mullidi 11141	. . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π))
21		pipos 26441	. . . . . . . . . 10 ⊢ 0 < π
22	12, 13, 21	ltleii 11260	. . . . . . . . 9 ⊢ 0 ≤ π
23		volioo 25554	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0))
24	12, 13, 22, 23	mp3an 1469	. . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0)
25	13	recni 11150	. . . . . . . . 9 ⊢ π ∈ ℂ
26	25	subid1i 11457	. . . . . . . 8 ⊢ (π − 0) = π
27	24, 26	eqtri 2762	. . . . . . 7 ⊢ (vol‘(0(,)π)) = π
28	20, 27	eqtri 2762	. . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π
29	18, 28	eqtri 2762	. . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π
30	10, 29	eqtrdi 2790	. . . 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 3453	. . . 4 ⊢ π ∈ V
33	30, 31, 32	fvmpt 6935	. . 3 ⊢ (0 ∈ ℕ₀ → (𝐼‘0) = π)
34	1, 33	ax-mp 5	. 2 ⊢ (𝐼‘0) = π
35		1nn0 12444	. . . 4 ⊢ 1 ∈ ℕ₀
36		simpl 483	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1)
37	36	oveq2d 7372	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1))
38	6	adantl 482	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
39	38	exp1d 14094	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥))
40	37, 39	eqtrd 2774	. . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥))
41	40	itgeq2dv 25767	. . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
42		itgex 25755	. . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V
43	41, 31, 42	fvmpt 6935	. . . 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 46395	. . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
46	44, 45	eqtri 2762	. 2 ⊢ (𝐼‘1) = 2
47		id 22	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘2))
48	31, 47	itgsinexp 46398	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
49	34, 46, 48	3pm3.2i 1346	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 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 · cmul 11034 ≤ cle 11171 − cmin 11368 / cdiv 11798 2c2 12227 ℕ₀cn0 12428 ℤ_≥cuz 12779 (,)cioo 13289 ↑cexp 14014 sincsin 16019 πcpi 16022 volcvol 25448 ∫citg 25603

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-symdif 4181 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-ovol 25449 df-vol 25450 df-mbf 25604 df-itg1 25605 df-itg2 25606 df-ibl 25607 df-itg 25608 df-0p 25655 df-limc 25851 df-dv 25852

This theorem is referenced by: wallispilem3 46510 wallispilem4 46511

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