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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 11515 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | bpolyval 14986 | . . 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 670 | . 2 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) |
4 | exp1 13073 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑1) = 𝑋) | |
5 | 1m1e0 11295 | . . . . . 6 ⊢ (1 − 1) = 0 | |
6 | 5 | oveq2i 6807 | . . . . 5 ⊢ (0...(1 − 1)) = (0...0) |
7 | 6 | sumeq1i 14636 | . . . 4 ⊢ Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) |
8 | 0z 11595 | . . . . . 6 ⊢ 0 ∈ ℤ | |
9 | bpoly0 14987 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1) | |
10 | 9 | oveq1d 6811 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 2) = (1 / 2)) |
11 | 10 | oveq2d 6812 | . . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 · (1 / 2))) |
12 | halfcn 11454 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ | |
13 | 12 | mulid2i 10249 | . . . . . . . 8 ⊢ (1 · (1 / 2)) = (1 / 2) |
14 | 11, 13 | syl6eq 2821 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 / 2)) |
15 | 14, 12 | syl6eqel 2858 | . . . . . 6 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) |
16 | oveq2 6804 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (1C𝑘) = (1C0)) | |
17 | bcn0 13301 | . . . . . . . . . 10 ⊢ (1 ∈ ℕ0 → (1C0) = 1) | |
18 | 1, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ (1C0) = 1 |
19 | 16, 18 | syl6eq 2821 | . . . . . . . 8 ⊢ (𝑘 = 0 → (1C𝑘) = 1) |
20 | oveq1 6803 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) | |
21 | oveq2 6804 | . . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (1 − 𝑘) = (1 − 0)) | |
22 | 1m0e1 11337 | . . . . . . . . . . . 12 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2821 | . . . . . . . . . . 11 ⊢ (𝑘 = 0 → (1 − 𝑘) = 1) |
24 | 23 | oveq1d 6811 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = (1 + 1)) |
25 | df-2 11285 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
26 | 24, 25 | syl6eqr 2823 | . . . . . . . . 9 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = 2) |
27 | 20, 26 | oveq12d 6814 | . . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 2)) |
28 | 19, 27 | oveq12d 6814 | . . . . . . 7 ⊢ (𝑘 = 0 → ((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
29 | 28 | fsum1 14684 | . . . . . 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 575 | . . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
31 | 30, 14 | eqtrd 2805 | . . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
32 | 7, 31 | syl5eq 2817 | . . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
33 | 4, 32 | oveq12d 6814 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)))) = (𝑋 − (1 / 2))) |
34 | 3, 33 | eqtrd 2805 | 1 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 (class class class)co 6796 ℂcc 10140 0cc0 10142 1c1 10143 + caddc 10145 · cmul 10147 − cmin 10472 / cdiv 10890 2c2 11276 ℕ0cn0 11499 ℤcz 11584 ...cfz 12533 ↑cexp 13067 Ccbc 13293 Σcsu 14624 BernPoly cbp 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8508 df-oi 8575 df-card 8969 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-bpoly 14984 |
This theorem is referenced by: bpoly2 14994 bpoly3 14995 bpoly4 14996 |
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