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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 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑛 = 0 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) | |
3 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑0)) |
4 | ioosscn 13446 | . . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ | |
5 | 4 | sseli 3991 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ) |
6 | 5 | sincld 16163 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ) |
7 | 6 | adantl 481 | . . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
8 | 7 | exp0d 14177 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1) |
9 | 3, 8 | eqtrd 2775 | . . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1) |
10 | 9 | itgeq2dv 25832 | . . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥) |
11 | ioombl 25614 | . . . . . . 7 ⊢ (0(,)π) ∈ dom vol | |
12 | 0re 11261 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
13 | pire 26515 | . . . . . . . 8 ⊢ π ∈ ℝ | |
14 | ioovolcl 25619 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ) | |
15 | 12, 13, 14 | mp2an 692 | . . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ |
16 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | itgconst 25869 | . . . . . . 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 1460 | . . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π))) |
19 | 15 | recni 11273 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ |
20 | 19 | mullidi 11264 | . . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π)) |
21 | pipos 26517 | . . . . . . . . . 10 ⊢ 0 < π | |
22 | 12, 13, 21 | ltleii 11382 | . . . . . . . . 9 ⊢ 0 ≤ π |
23 | volioo 25618 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0)) | |
24 | 12, 13, 22, 23 | mp3an 1460 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0) |
25 | 13 | recni 11273 | . . . . . . . . 9 ⊢ π ∈ ℂ |
26 | 25 | subid1i 11579 | . . . . . . . 8 ⊢ (π − 0) = π |
27 | 24, 26 | eqtri 2763 | . . . . . . 7 ⊢ (vol‘(0(,)π)) = π |
28 | 20, 27 | eqtri 2763 | . . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π |
29 | 18, 28 | eqtri 2763 | . . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π |
30 | 10, 29 | eqtrdi 2791 | . . . 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 3501 | . . . 4 ⊢ π ∈ V |
33 | 30, 31, 32 | fvmpt 7016 | . . 3 ⊢ (0 ∈ ℕ0 → (𝐼‘0) = π) |
34 | 1, 33 | ax-mp 5 | . 2 ⊢ (𝐼‘0) = π |
35 | 1nn0 12540 | . . . 4 ⊢ 1 ∈ ℕ0 | |
36 | simpl 482 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → 𝑛 = 1) | |
37 | 36 | oveq2d 7447 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑1)) |
38 | 6 | adantl 481 | . . . . . . . 8 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
39 | 38 | exp1d 14178 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥)) |
40 | 37, 39 | eqtrd 2775 | . . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥)) |
41 | 40 | itgeq2dv 25832 | . . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥) |
42 | itgex 25820 | . . . . 5 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 ∈ V | |
43 | 41, 31, 42 | fvmpt 7016 | . . . 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 45908 | . . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2 | |
46 | 44, 45 | eqtri 2763 | . 2 ⊢ (𝐼‘1) = 2 |
47 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘2)) | |
48 | 31, 47 | itgsinexp 45911 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))) |
49 | 34, 46, 48 | 3pm3.2i 1338 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 ≤ cle 11294 − cmin 11490 / cdiv 11918 2c2 12319 ℕ0cn0 12524 ℤ≥cuz 12876 (,)cioo 13384 ↑cexp 14099 sincsin 16096 πcpi 16099 volcvol 25512 ∫citg 25667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-limc 25916 df-dv 25917 |
This theorem is referenced by: wallispilem3 46023 wallispilem4 46024 |
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