wallispilem1 - Mathbox for Glauco Siliprandi

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Theorem wallispilem1 43496

Description: 𝐼 is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
wallispilem1.1	⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)
wallispilem1.2	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
wallispilem1	⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁))

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

**Proof of Theorem wallispilem1**
Step	Hyp	Ref	Expression
1		0re 10908	. . . . 5 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
3		pire 25520	. . . . 5 ⊢ π ∈ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
5		wallispilem1.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		peano2nn0 12203	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
8		iblioosinexp 43384	. . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿¹)
9	2, 4, 7, 8	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿¹)
10		iblioosinexp 43384	. . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
11	2, 4, 5, 10	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
12		elioore 13038	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℝ)
13	12	resincld 15780	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℝ)
14	13	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℝ)
15	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ ℕ₀)
16	14, 15	reexpcld 13809	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ∈ ℝ)
17	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
18	14, 17	reexpcld 13809	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℝ)
19	5	nn0zd 12353	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20		uzid 12526	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
22		peano2uz 12570	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑁) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
23	21, 22	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
24	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
25	13, 1	jctil 519	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ))
26		sinq12gt0 25569	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 0 < (sin‘𝑥))
27		ltle 10994	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ) → (0 < (sin‘𝑥) → 0 ≤ (sin‘𝑥)))
28	25, 26, 27	sylc 65	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → 0 ≤ (sin‘𝑥))
29	28	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 0 ≤ (sin‘𝑥))
30		sinbnd 15817	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (-1 ≤ (sin‘𝑥) ∧ (sin‘𝑥) ≤ 1))
31	12, 30	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (-1 ≤ (sin‘𝑥) ∧ (sin‘𝑥) ≤ 1))
32	31	simprd 495	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ≤ 1)
33	32	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ≤ 1)
34	14, 17, 24, 29, 33	leexp2rd 13900	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ≤ ((sin‘𝑥)↑𝑁))
35	9, 11, 16, 18, 34	itgle 24879	. 2 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ≤ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
36		oveq2 7263	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1)))
37	36	adantr 480	. . . . 5 ⊢ ((𝑛 = (𝑁 + 1) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1)))
38	37	itgeq2dv 24851	. . . 4 ⊢ (𝑛 = (𝑁 + 1) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥)
39		wallispilem1.1	. . . 4 ⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)
40		itgex 24840	. . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ∈ V
41	38, 39, 40	fvmpt 6857	. . 3 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝐼‘(𝑁 + 1)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥)
42	7, 41	syl 17	. 2 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥)
43		oveq2 7263	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
44	43	adantr 480	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
45	44	itgeq2dv 24851	. . . 4 ⊢ (𝑛 = 𝑁 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
46		itgex 24840	. . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V
47	45, 39, 46	fvmpt 6857	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
48	5, 47	syl 17	. 2 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
49	35, 42, 48	3brtr4d 5102	1 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 -cneg 11136 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 (,)cioo 13008 ↑cexp 13710 sincsin 15701 πcpi 15704 𝐿¹cibl 24686 ∫citg 24687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936

This theorem is referenced by: wallispilem5 43500

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