wallispilem1 - Mathbox for Glauco Siliprandi

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Theorem wallispilem1 46705

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 11210	. . . . 5 ⊢ 0 ∈ ℝ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
3		pire 26585	. . . . 5 ⊢ π ∈ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
5		wallispilem1.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		peano2nn0 12544	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7	5, 6	syl 18	. . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
8		iblioosinexp 46593	. . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿¹)
9	2, 4, 7, 8	syl3anc 1396	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿¹)
10		iblioosinexp 46593	. . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
11	2, 4, 5, 10	syl3anc 1396	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
12		elioore 13402	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℝ)
13	12	resincld 16199	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℝ)
14	13	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℝ)
15	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ ℕ₀)
16	14, 15	reexpcld 14199	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ∈ ℝ)
17	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
18	14, 17	reexpcld 14199	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℝ)
19	5	nn0zd 12616	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
20		uzid 12877	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
21	19, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
22		peano2uz 12925	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑁) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
23	21, 22	syl 18	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
24	23	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
25	13, 1	jctil 528	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ))
26		sinq12gt0 26638	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 0 < (sin‘𝑥))
27		ltle 11298	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ) → (0 < (sin‘𝑥) → 0 ≤ (sin‘𝑥)))
28	25, 26, 27	sylc 66	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → 0 ≤ (sin‘𝑥))
29	28	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 0 ≤ (sin‘𝑥))
30		sinbnd 16236	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (-1 ≤ (sin‘𝑥) ∧ (sin‘𝑥) ≤ 1))
31	12, 30	syl 18	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (-1 ≤ (sin‘𝑥) ∧ (sin‘𝑥) ≤ 1))
32	31	simprd 500	. . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ≤ 1)
33	32	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ≤ 1)
34	14, 17, 24, 29, 33	leexp2rd 14291	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ≤ ((sin‘𝑥)↑𝑁))
35	9, 11, 16, 18, 34	itgle 25938	. 2 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ≤ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
36		oveq2 7419	. . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1)))
37	36	adantr 485	. . . . 5 ⊢ ((𝑛 = (𝑁 + 1) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1)))
38	37	itgeq2dv 25910	. . . 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 25898	. . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ∈ V
41	38, 39, 40	fvmpt 6990	. . 3 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝐼‘(𝑁 + 1)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥)
42	7, 41	syl 18	. 2 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥)
43		oveq2 7419	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
44	43	adantr 485	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
45	44	itgeq2dv 25910	. . . 4 ⊢ (𝑛 = 𝑁 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
46		itgex 25898	. . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V
47	45, 39, 46	fvmpt 6990	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
48	5, 47	syl 18	. 2 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
49	35, 42, 48	3brtr4d 5147	1 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 class class class wbr 5113 ↦ cmpt 5196 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 -cneg 11442 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 (,)cioo 13372 ↑cexp 14097 sincsin 16117 πcpi 16120 𝐿¹cibl 25745 ∫citg 25746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cc 10419 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-ofr 7676 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15104 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-ef 16121 df-sin 16123 df-cos 16124 df-pi 16126 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-ovol 25592 df-vol 25593 df-mbf 25747 df-itg1 25748 df-itg2 25749 df-ibl 25750 df-itg 25751 df-0p 25798 df-limc 25994 df-dv 25995

This theorem is referenced by: wallispilem5 46709

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