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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 10643 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) |
3 | pire 25044 | . . . . 5 ⊢ π ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → π ∈ ℝ) |
5 | wallispilem1.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | peano2nn0 11938 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0) |
8 | iblioosinexp 42258 | . . . 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 42258 | . . . 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 12769 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℝ) | |
13 | 12 | resincld 15496 | . . . . 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 13528 | . . 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 13528 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℝ) |
19 | 5 | nn0zd 12086 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
20 | uzid 12259 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
22 | peano2uz 12302 | . . . . . 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 25093 | . . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → 0 < (sin‘𝑥)) | |
27 | ltle 10729 | . . . . . 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 15533 | . . . . . . 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 13619 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 + 1)) ≤ ((sin‘𝑥)↑𝑁)) |
35 | 9, 11, 16, 18, 34 | itgle 24410 | . 2 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ≤ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
36 | oveq2 7164 | . . . . . 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 24382 | . . . 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 24371 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 + 1)) d𝑥 ∈ V | |
41 | 38, 39, 40 | fvmpt 6768 | . . 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 7164 | . . . . . 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 24382 | . . . 4 ⊢ (𝑛 = 𝑁 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥) |
46 | itgex 24371 | . . . 4 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V | |
47 | 45, 39, 46 | fvmpt 6768 | . . 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 5098 | 1 ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 -cneg 10871 ℕ0cn0 11898 ℤcz 11982 ℤ≥cuz 12244 (,)cioo 12739 ↑cexp 13430 sincsin 15417 πcpi 15420 𝐿1cibl 24218 ∫citg 24219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 df-itg2 24222 df-ibl 24223 df-itg 24224 df-0p 24271 df-limc 24464 df-dv 24465 |
This theorem is referenced by: wallispilem5 42374 |
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