Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wallispilem2 | Structured version Visualization version GIF version |
Description: A first set of properties for the sequence 𝐼 that will be used in the proof of the Wallis product formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
wallispilem2.1 | ⊢ 𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥) |
Ref | Expression |
---|---|
wallispilem2 | ⊢ ((𝐼‘0) = π ∧ (𝐼‘1) = 2 ∧ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11307 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | oveq2 6658 | . . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) | |
3 | 2 | adantr 481 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) |
4 | ioosscn 39716 | . . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ | |
5 | 4 | sseli 3599 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ) |
6 | 5 | sincld 14860 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ) |
7 | 6 | adantl 482 | . . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
8 | 7 | exp0d 13002 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1) |
9 | 3, 8 | eqtrd 2656 | . . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1) |
10 | 9 | itgeq2dv 23548 | . . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥) |
11 | ioombl 23333 | . . . . . . 7 ⊢ (0(,)π) ∈ dom vol | |
12 | 0re 10040 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
13 | pire 24210 | . . . . . . . 8 ⊢ π ∈ ℝ | |
14 | ioovolcl 23338 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ) | |
15 | 12, 13, 14 | mp2an 708 | . . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ |
16 | ax-1cn 9994 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | itgconst 23585 | . . . . . . 7 ⊢ (((0(,)π) ∈ dom vol ∧ (vol‘(0(,)π)) ∈ ℝ ∧ 1 ∈ ℂ) → ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π)))) | |
18 | 11, 15, 16, 17 | mp3an 1424 | . . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π))) |
19 | 15 | recni 10052 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ |
20 | 19 | mulid2i 10043 | . . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π)) |
21 | pipos 24212 | . . . . . . . . . 10 ⊢ 0 < π | |
22 | 12, 13, 21 | ltleii 10160 | . . . . . . . . 9 ⊢ 0 ≤ π |
23 | volioo 23337 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0)) | |
24 | 12, 13, 22, 23 | mp3an 1424 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0) |
25 | 13 | recni 10052 | . . . . . . . . 9 ⊢ π ∈ ℂ |
26 | 25 | subid1i 10353 | . . . . . . . 8 ⊢ (π − 0) = π |
27 | 24, 26 | eqtri 2644 | . . . . . . 7 ⊢ (vol‘(0(,)π)) = π |
28 | 20, 27 | eqtri 2644 | . . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π |
29 | 18, 28 | eqtri 2644 | . . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π |
30 | 10, 29 | syl6eq 2672 | . . . 4 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = π) |
31 | wallispilem2.1 | . . . 4 ⊢ 𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥) | |
32 | 13 | elexi 3213 | . . . 4 ⊢ π ∈ V |
33 | 30, 31, 32 | fvmpt 6282 | . . 3 ⊢ (0 ∈ ℕ0 → (𝐼‘0) = π) |
34 | 1, 33 | ax-mp 5 | . 2 ⊢ (𝐼‘0) = π |
35 | 1nn0 11308 | . . . 4 ⊢ 1 ∈ ℕ0 | |
36 | simpl 473 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1) | |
37 | 36 | oveq2d 6666 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1)) |
38 | 6 | adantl 482 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
39 | 38 | exp1d 13003 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥)) |
40 | 37, 39 | eqtrd 2656 | . . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥)) |
41 | 40 | itgeq2dv 23548 | . . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥) |
42 | itgex 23537 | . . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V | |
43 | 41, 31, 42 | fvmpt 6282 | . . . 4 ⊢ (1 ∈ ℕ0 → (𝐼‘1) = ∫(0(,)π)(sin‘𝑥) d𝑥) |
44 | 35, 43 | ax-mp 5 | . . 3 ⊢ (𝐼‘1) = ∫(0(,)π)(sin‘𝑥) d𝑥 |
45 | itgsin0pi 40167 | . . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2 | |
46 | 44, 45 | eqtri 2644 | . 2 ⊢ (𝐼‘1) = 2 |
47 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘2)) | |
48 | 31, 47 | itgsinexp 40170 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))) |
49 | 34, 46, 48 | 3pm3.2i 1239 | 1 ⊢ ((𝐼‘0) = π ∧ (𝐼‘1) = 2 ∧ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 ℕ0cn0 11292 ℤ≥cuz 11687 (,)cioo 12175 ↑cexp 12860 sincsin 14794 πcpi 14797 volcvol 23232 ∫citg 23387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-limc 23630 df-dv 23631 |
This theorem is referenced by: wallispilem3 40284 wallispilem4 40285 |
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