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Mirrors > Home > MPE Home > Th. List > bpoly1 | Structured version Visualization version GIF version |
Description: The value of the Bernoulli polynomials at one. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
bpoly1 | ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11660 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | bpolyval 15182 | . . 3 ⊢ ((1 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) | |
3 | 1, 2 | mpan 680 | . 2 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) |
4 | exp1 13184 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑1) = 𝑋) | |
5 | 1m1e0 11447 | . . . . . 6 ⊢ (1 − 1) = 0 | |
6 | 5 | oveq2i 6933 | . . . . 5 ⊢ (0...(1 − 1)) = (0...0) |
7 | 6 | sumeq1i 14836 | . . . 4 ⊢ Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) |
8 | 0z 11739 | . . . . . 6 ⊢ 0 ∈ ℤ | |
9 | bpoly0 15183 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1) | |
10 | 9 | oveq1d 6937 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 2) = (1 / 2)) |
11 | 10 | oveq2d 6938 | . . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 · (1 / 2))) |
12 | halfcn 11597 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ | |
13 | 12 | mulid2i 10382 | . . . . . . . 8 ⊢ (1 · (1 / 2)) = (1 / 2) |
14 | 11, 13 | syl6eq 2830 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 / 2)) |
15 | 14, 12 | syl6eqel 2867 | . . . . . 6 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) |
16 | oveq2 6930 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (1C𝑘) = (1C0)) | |
17 | bcn0 13415 | . . . . . . . . . 10 ⊢ (1 ∈ ℕ0 → (1C0) = 1) | |
18 | 1, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ (1C0) = 1 |
19 | 16, 18 | syl6eq 2830 | . . . . . . . 8 ⊢ (𝑘 = 0 → (1C𝑘) = 1) |
20 | oveq1 6929 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) | |
21 | oveq2 6930 | . . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (1 − 𝑘) = (1 − 0)) | |
22 | 1m0e1 11503 | . . . . . . . . . . . 12 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2830 | . . . . . . . . . . 11 ⊢ (𝑘 = 0 → (1 − 𝑘) = 1) |
24 | 23 | oveq1d 6937 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = (1 + 1)) |
25 | df-2 11438 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
26 | 24, 25 | syl6eqr 2832 | . . . . . . . . 9 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = 2) |
27 | 20, 26 | oveq12d 6940 | . . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 2)) |
28 | 19, 27 | oveq12d 6940 | . . . . . . 7 ⊢ (𝑘 = 0 → ((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
29 | 28 | fsum1 14883 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
30 | 8, 15, 29 | sylancr 581 | . . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
31 | 30, 14 | eqtrd 2814 | . . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
32 | 7, 31 | syl5eq 2826 | . . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
33 | 4, 32 | oveq12d 6940 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)))) = (𝑋 − (1 / 2))) |
34 | 3, 33 | eqtrd 2814 | 1 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 − cmin 10606 / cdiv 11032 2c2 11430 ℕ0cn0 11642 ℤcz 11728 ...cfz 12643 ↑cexp 13178 Ccbc 13407 Σcsu 14824 BernPoly cbp 15179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-bpoly 15180 |
This theorem is referenced by: bpoly2 15190 bpoly3 15191 bpoly4 15192 |
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