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Mirrors > Home > MPE Home > Th. List > Mathboxes > wallispilem1 | Structured version Visualization version GIF version |
Description: 𝐼 is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
wallispilem1.1 | ⊢ 𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥) |
wallispilem1.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
wallispilem1 | ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10646 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) |
3 | pire 25047 | . . . . 5 ⊢ π ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → π ∈ ℝ) |
5 | wallispilem1.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | peano2nn0 11940 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
8 | iblioosinexp 42244 | . . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ0) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿1) | |
9 | 2, 4, 7, 8 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿1) |
10 | iblioosinexp 42244 | . . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1) | |
11 | 2, 4, 5, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1) |
12 | elioore 12771 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℝ) | |
13 | 12 | resincld 15499 | . . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℝ) |
14 | 13 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℝ) |
15 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ ℕ0) |
16 | 14, 15 | reexpcld 13530 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ∈ ℝ) |
17 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ0) |
18 | 14, 17 | reexpcld 13530 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℝ) |
19 | 5 | nn0zd 12088 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | uzid 12261 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
22 | peano2uz 12304 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) |
24 | 23 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) |
25 | 13, 1 | jctil 522 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ)) |
26 | sinq12gt0 25096 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 0 < (sin‘𝑥)) | |
27 | ltle 10732 | . . . . . 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 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 0 ≤ (sin‘𝑥)) |
30 | sinbnd 15536 | . . . . . . 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 498 | . . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ≤ 1) |
33 | 32 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ≤ 1) |
34 | 14, 17, 24, 29, 33 | leexp2rd 13621 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ≤ ((sin‘𝑥)↑𝑁)) |
35 | 9, 11, 16, 18, 34 | itgle 24413 | . 2 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ≤ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
36 | oveq2 7167 | . . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1))) | |
37 | 36 | adantr 483 | . . . . 5 ⊢ ((𝑛 = (𝑁 + 1) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1))) |
38 | 37 | itgeq2dv 24385 | . . . 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 24374 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ∈ V | |
41 | 38, 39, 40 | fvmpt 6771 | . . 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 7167 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁)) | |
44 | 43 | adantr 483 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁)) |
45 | 44 | itgeq2dv 24385 | . . . 4 ⊢ (𝑛 = 𝑁 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
46 | itgex 24374 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V | |
47 | 45, 39, 46 | fvmpt 6771 | . . 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 5101 | 1 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 < clt 10678 ≤ cle 10679 -cneg 10874 ℕ0cn0 11900 ℤcz 11984 ℤ≥cuz 12246 (,)cioo 12741 ↑cexp 13432 sincsin 15420 πcpi 15423 𝐿1cibl 24221 ∫citg 24222 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cc 9860 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-cmp 21998 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-limc 24467 df-dv 24468 |
This theorem is referenced by: wallispilem5 42361 |
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