wallispilem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > wallispilem2		Structured version Visualization version GIF version

Theorem wallispilem2 41782

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 11718	. . 3 ⊢ 0 ∈ ℕ₀
2		oveq2 6978	. . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
3	2	adantr 473	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0))
4		ioosscn 41200	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
5	4	sseli 3848	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
6	5	sincld 15337	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
7	6	adantl 474	. . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
8	7	exp0d 13313	. . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1)
9	3, 8	eqtrd 2808	. . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1)
10	9	itgeq2dv 24079	. . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥)
11		ioombl 23863	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
12		0re 10435	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		pire 24741	. . . . . . . 8 ⊢ π ∈ ℝ
14		ioovolcl 23868	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ)
15	12, 13, 14	mp2an 679	. . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ
16		ax-1cn 10387	. . . . . . 7 ⊢ 1 ∈ ℂ
17		itgconst 24116	. . . . . . 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 1440	. . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π)))
19	15	recni 10448	. . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ
20	19	mulid2i 10439	. . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π))
21		pipos 24743	. . . . . . . . . 10 ⊢ 0 < π
22	12, 13, 21	ltleii 10557	. . . . . . . . 9 ⊢ 0 ≤ π
23		volioo 23867	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0))
24	12, 13, 22, 23	mp3an 1440	. . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0)
25	13	recni 10448	. . . . . . . . 9 ⊢ π ∈ ℂ
26	25	subid1i 10753	. . . . . . . 8 ⊢ (π − 0) = π
27	24, 26	eqtri 2796	. . . . . . 7 ⊢ (vol‘(0(,)π)) = π
28	20, 27	eqtri 2796	. . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π
29	18, 28	eqtri 2796	. . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π
30	10, 29	syl6eq 2824	. . . 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 3428	. . . 4 ⊢ π ∈ V
33	30, 31, 32	fvmpt 6589	. . 3 ⊢ (0 ∈ ℕ₀ → (𝐼‘0) = π)
34	1, 33	ax-mp 5	. 2 ⊢ (𝐼‘0) = π
35		1nn0 11719	. . . 4 ⊢ 1 ∈ ℕ₀
36		simpl 475	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1)
37	36	oveq2d 6986	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1))
38	6	adantl 474	. . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
39	38	exp1d 13314	. . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥))
40	37, 39	eqtrd 2808	. . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥))
41	40	itgeq2dv 24079	. . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
42		itgex 24068	. . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V
43	41, 31, 42	fvmpt 6589	. . . 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 41667	. . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
46	44, 45	eqtri 2796	. 2 ⊢ (𝐼‘1) = 2
47		id 22	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ (ℤ_≥‘2))
48	31, 47	itgsinexp 41670	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
49	34, 46, 48	3pm3.2i 1319	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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5401 ‘cfv 6182 (class class class)co 6970 ℂcc 10327 ℝcr 10328 0cc0 10329 1c1 10330 · cmul 10334 ≤ cle 10469 − cmin 10664 / cdiv 11092 2c2 11489 ℕ₀cn0 11701 ℤ_≥cuz 12052 (,)cioo 12548 ↑cexp 13238 sincsin 15271 πcpi 15274 volcvol 23761 ∫citg 23916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cc 9649 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-symdif 4100 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-disj 4892 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-omul 7904 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-dju 9118 df-card 9156 df-acn 9159 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ioc 12553 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-mod 13047 df-seq 13179 df-exp 13239 df-fac 13443 df-bc 13472 df-hash 13500 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-cmp 21693 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-ovol 23762 df-vol 23763 df-mbf 23917 df-itg1 23918 df-itg2 23919 df-ibl 23920 df-itg 23921 df-0p 23968 df-limc 24161 df-dv 24162

This theorem is referenced by: wallispilem3 41783 wallispilem4 41784

W3C validator