| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| 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 12541 | . . 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 13449 | . . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ | |
| 5 | 4 | sseli 3979 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ) |
| 6 | 5 | sincld 16166 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ) |
| 7 | 6 | adantl 481 | . . . . . . . 8 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ) |
| 8 | 7 | exp0d 14180 | . . . . . . 7 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑0) = 1) |
| 9 | 3, 8 | eqtrd 2777 | . . . . . 6 ⊢ ((𝑛 = 0 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = 1) |
| 10 | 9 | itgeq2dv 25817 | . . . . 5 ⊢ (𝑛 = 0 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)1 d𝑥) |
| 11 | ioombl 25600 | . . . . . . 7 ⊢ (0(,)π) ∈ dom vol | |
| 12 | 0re 11263 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
| 13 | pire 26500 | . . . . . . . 8 ⊢ π ∈ ℝ | |
| 14 | ioovolcl 25605 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (vol‘(0(,)π)) ∈ ℝ) | |
| 15 | 12, 13, 14 | mp2an 692 | . . . . . . 7 ⊢ (vol‘(0(,)π)) ∈ ℝ |
| 16 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 17 | itgconst 25854 | . . . . . . 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 1463 | . . . . . 6 ⊢ ∫(0(,)π)1 d𝑥 = (1 · (vol‘(0(,)π))) |
| 19 | 15 | recni 11275 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) ∈ ℂ |
| 20 | 19 | mullidi 11266 | . . . . . . 7 ⊢ (1 · (vol‘(0(,)π))) = (vol‘(0(,)π)) |
| 21 | pipos 26502 | . . . . . . . . . 10 ⊢ 0 < π | |
| 22 | 12, 13, 21 | ltleii 11384 | . . . . . . . . 9 ⊢ 0 ≤ π |
| 23 | volioo 25604 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ 0 ≤ π) → (vol‘(0(,)π)) = (π − 0)) | |
| 24 | 12, 13, 22, 23 | mp3an 1463 | . . . . . . . 8 ⊢ (vol‘(0(,)π)) = (π − 0) |
| 25 | 13 | recni 11275 | . . . . . . . . 9 ⊢ π ∈ ℂ |
| 26 | 25 | subid1i 11581 | . . . . . . . 8 ⊢ (π − 0) = π |
| 27 | 24, 26 | eqtri 2765 | . . . . . . 7 ⊢ (vol‘(0(,)π)) = π |
| 28 | 20, 27 | eqtri 2765 | . . . . . 6 ⊢ (1 · (vol‘(0(,)π))) = π |
| 29 | 18, 28 | eqtri 2765 | . . . . 5 ⊢ ∫(0(,)π)1 d𝑥 = π |
| 30 | 10, 29 | eqtrdi 2793 | . . . 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 3503 | . . . 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 12542 | . . . 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 14181 | . . . . . . 7 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑1) = (sin‘𝑥)) |
| 40 | 37, 39 | eqtrd 2777 | . . . . . 6 ⊢ ((𝑛 = 1 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = (sin‘𝑥)) |
| 41 | 40 | itgeq2dv 25817 | . . . . 5 ⊢ (𝑛 = 1 → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥) |
| 42 | itgex 25805 | . . . . 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 45967 | . . 3 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2 | |
| 46 | 44, 45 | eqtri 2765 | . 2 ⊢ (𝐼‘1) = 2 |
| 47 | id 22 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ (ℤ≥‘2)) | |
| 48 | 31, 47 | itgsinexp 45970 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))) |
| 49 | 34, 46, 48 | 3pm3.2i 1340 | 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 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 ℕ0cn0 12526 ℤ≥cuz 12878 (,)cioo 13387 ↑cexp 14102 sincsin 16099 πcpi 16102 volcvol 25498 ∫citg 25653 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902 |
| This theorem is referenced by: wallispilem3 46082 wallispilem4 46083 |
| Copyright terms: Public domain | W3C validator |