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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 12534 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | oveq2 7431 | . . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) | |
3 | 2 | adantr 479 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) |
4 | ioosscn 13435 | . . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ | |
5 | 4 | sseli 3974 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ) |
6 | 5 | sincld 16127 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ) |
7 | 6 | adantl 480 | . . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
8 | 7 | exp0d 14154 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1) |
9 | 3, 8 | eqtrd 2765 | . . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1) |
10 | 9 | itgeq2dv 25794 | . . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥) |
11 | ioombl 25577 | . . . . . . 7 ⊢ (0(,)π) ∈ dom vol | |
12 | 0re 11262 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
13 | pire 26478 | . . . . . . . 8 ⊢ π ∈ ℝ | |
14 | ioovolcl 25582 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ) | |
15 | 12, 13, 14 | mp2an 690 | . . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ |
16 | ax-1cn 11212 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | itgconst 25831 | . . . . . . 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 1457 | . . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π))) |
19 | 15 | recni 11274 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ |
20 | 19 | mullidi 11265 | . . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π)) |
21 | pipos 26480 | . . . . . . . . . 10 ⊢ 0 < π | |
22 | 12, 13, 21 | ltleii 11383 | . . . . . . . . 9 ⊢ 0 ≤ π |
23 | volioo 25581 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0)) | |
24 | 12, 13, 22, 23 | mp3an 1457 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0) |
25 | 13 | recni 11274 | . . . . . . . . 9 ⊢ π ∈ ℂ |
26 | 25 | subid1i 11578 | . . . . . . . 8 ⊢ (π − 0) = π |
27 | 24, 26 | eqtri 2753 | . . . . . . 7 ⊢ (vol‘(0(,)π)) = π |
28 | 20, 27 | eqtri 2753 | . . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π |
29 | 18, 28 | eqtri 2753 | . . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π |
30 | 10, 29 | eqtrdi 2781 | . . . 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 3483 | . . . 4 ⊢ π ∈ V |
33 | 30, 31, 32 | fvmpt 7008 | . . 3 ⊢ (0 ∈ ℕ0 → (𝐼‘0) = π) |
34 | 1, 33 | ax-mp 5 | . 2 ⊢ (𝐼‘0) = π |
35 | 1nn0 12535 | . . . 4 ⊢ 1 ∈ ℕ0 | |
36 | simpl 481 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1) | |
37 | 36 | oveq2d 7439 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1)) |
38 | 6 | adantl 480 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
39 | 38 | exp1d 14155 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥)) |
40 | 37, 39 | eqtrd 2765 | . . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥)) |
41 | 40 | itgeq2dv 25794 | . . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥) |
42 | itgex 25783 | . . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V | |
43 | 41, 31, 42 | fvmpt 7008 | . . . 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 45510 | . . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2 | |
46 | 44, 45 | eqtri 2753 | . 2 ⊢ (𝐼‘1) = 2 |
47 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘2)) | |
48 | 31, 47 | itgsinexp 45513 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))) |
49 | 34, 46, 48 | 3pm3.2i 1336 | 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5681 ‘cfv 6553 (class class class)co 7423 ℂcc 11152 ℝcr 11153 0cc0 11154 1c1 11155 · cmul 11159 ≤ cle 11295 − cmin 11490 / cdiv 11917 2c2 12314 ℕ0cn0 12519 ℤ≥cuz 12869 (,)cioo 13373 ↑cexp 14076 sincsin 16060 πcpi 16063 volcvol 25475 ∫citg 25630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cc 10474 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-symdif 4243 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-ofr 7690 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8856 df-pm 8857 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-dju 9940 df-card 9978 df-acn 9981 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-fl 13807 df-mod 13885 df-seq 14017 df-exp 14077 df-fac 14286 df-bc 14315 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-sin 16066 df-cos 16067 df-pi 16069 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19057 df-cntz 19306 df-cmn 19775 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-fbas 21332 df-fg 21333 df-cnfld 21336 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-cmp 23374 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24309 df-ms 24310 df-tms 24311 df-cncf 24881 df-ovol 25476 df-vol 25477 df-mbf 25631 df-itg1 25632 df-itg2 25633 df-ibl 25634 df-itg 25635 df-0p 25682 df-limc 25878 df-dv 25879 |
This theorem is referenced by: wallispilem3 45625 wallispilem4 45626 |
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