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Mathbox for Glauco Siliprandi |
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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 10380 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) |
3 | pire 24659 | . . . . 5 ⊢ π ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → π ∈ ℝ) |
5 | wallispilem1.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | peano2nn0 11689 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
8 | iblioosinexp 41110 | . . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ0) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿1) | |
9 | 2, 4, 7, 8 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 + 1))) ∈ 𝐿1) |
10 | iblioosinexp 41110 | . . . 4 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1) | |
11 | 2, 4, 5, 10 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1) |
12 | elioore 12522 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℝ) | |
13 | 12 | resincld 15284 | . . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℝ) |
14 | 13 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℝ) |
15 | 7 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ ℕ0) |
16 | 14, 15 | reexpcld 13349 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ∈ ℝ) |
17 | 5 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ0) |
18 | 14, 17 | reexpcld 13349 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℝ) |
19 | 5 | nn0zd 11837 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | uzid 12012 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
22 | peano2uz 12052 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) |
24 | 23 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 + 1) ∈ (ℤ≥‘𝑁)) |
25 | 13, 1 | jctil 515 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → (0 ∈ ℝ ∧ (sin‘𝑥) ∈ ℝ)) |
26 | sinq12gt0 24708 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 0 < (sin‘𝑥)) | |
27 | ltle 10467 | . . . . . 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 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 0 ≤ (sin‘𝑥)) |
30 | sinbnd 15321 | . . . . . . 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 491 | . . . . 5 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ≤ 1) |
33 | 32 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ≤ 1) |
34 | 14, 17, 24, 29, 33 | leexp2rd 13369 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ≤ ((sin‘𝑥)↑𝑁)) |
35 | 9, 11, 16, 18, 34 | itgle 24024 | . 2 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ≤ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
36 | oveq2 6932 | . . . . . 6 ⊢ (𝑛 = (𝑁 + 1) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1))) | |
37 | 36 | adantr 474 | . . . . 5 ⊢ ((𝑛 = (𝑁 + 1) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 + 1))) |
38 | 37 | itgeq2dv 23996 | . . . 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 23985 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ∈ V | |
41 | 38, 39, 40 | fvmpt 6544 | . . 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 6932 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁)) | |
44 | 43 | adantr 474 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁)) |
45 | 44 | itgeq2dv 23996 | . . . 4 ⊢ (𝑛 = 𝑁 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
46 | itgex 23985 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V | |
47 | 45, 39, 46 | fvmpt 6544 | . . 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 4920 | 1 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 < clt 10413 ≤ cle 10414 -cneg 10609 ℕ0cn0 11647 ℤcz 11733 ℤ≥cuz 11997 (,)cioo 12492 ↑cexp 13183 sincsin 15205 πcpi 15208 𝐿1cibl 23832 ∫citg 23833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cc 9594 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-disj 4857 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-ofr 7177 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 df-fac 13385 df-bc 13414 df-hash 13442 df-shft 14220 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-limsup 14619 df-clim 14636 df-rlim 14637 df-sum 14834 df-ef 15209 df-sin 15211 df-cos 15212 df-pi 15214 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-fbas 20150 df-fg 20151 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-nei 21321 df-lp 21359 df-perf 21360 df-cn 21450 df-cnp 21451 df-haus 21538 df-cmp 21610 df-tx 21785 df-hmeo 21978 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-xms 22544 df-ms 22545 df-tms 22546 df-cncf 23100 df-ovol 23679 df-vol 23680 df-mbf 23834 df-itg1 23835 df-itg2 23836 df-ibl 23837 df-itg 23838 df-0p 23885 df-limc 24078 df-dv 24079 |
This theorem is referenced by: wallispilem5 41227 |
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