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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 11901 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | bpolyval 15391 | . . 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 686 | . 2 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) |
4 | exp1 13423 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑1) = 𝑋) | |
5 | 1m1e0 11697 | . . . . . 6 ⊢ (1 − 1) = 0 | |
6 | 5 | oveq2i 7156 | . . . . 5 ⊢ (0...(1 − 1)) = (0...0) |
7 | 6 | sumeq1i 15043 | . . . 4 ⊢ Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) |
8 | 0z 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
9 | bpoly0 15392 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1) | |
10 | 9 | oveq1d 7160 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 2) = (1 / 2)) |
11 | 10 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 · (1 / 2))) |
12 | halfcn 11840 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ | |
13 | 12 | mulid2i 10634 | . . . . . . . 8 ⊢ (1 · (1 / 2)) = (1 / 2) |
14 | 11, 13 | syl6eq 2869 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 / 2)) |
15 | 14, 12 | syl6eqel 2918 | . . . . . 6 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) |
16 | oveq2 7153 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (1C𝑘) = (1C0)) | |
17 | bcn0 13658 | . . . . . . . . . 10 ⊢ (1 ∈ ℕ0 → (1C0) = 1) | |
18 | 1, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ (1C0) = 1 |
19 | 16, 18 | syl6eq 2869 | . . . . . . . 8 ⊢ (𝑘 = 0 → (1C𝑘) = 1) |
20 | oveq1 7152 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) | |
21 | oveq2 7153 | . . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (1 − 𝑘) = (1 − 0)) | |
22 | 1m0e1 11746 | . . . . . . . . . . . 12 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2869 | . . . . . . . . . . 11 ⊢ (𝑘 = 0 → (1 − 𝑘) = 1) |
24 | 23 | oveq1d 7160 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = (1 + 1)) |
25 | df-2 11688 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
26 | 24, 25 | syl6eqr 2871 | . . . . . . . . 9 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = 2) |
27 | 20, 26 | oveq12d 7163 | . . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 2)) |
28 | 19, 27 | oveq12d 7163 | . . . . . . 7 ⊢ (𝑘 = 0 → ((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
29 | 28 | fsum1 15090 | . . . . . 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 587 | . . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
31 | 30, 14 | eqtrd 2853 | . . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
32 | 7, 31 | syl5eq 2865 | . . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
33 | 4, 32 | oveq12d 7163 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)))) = (𝑋 − (1 / 2))) |
34 | 3, 33 | eqtrd 2853 | 1 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 / cdiv 11285 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 ↑cexp 13417 Ccbc 13650 Σcsu 15030 BernPoly cbp 15388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-bpoly 15389 |
This theorem is referenced by: bpoly2 15399 bpoly3 15400 bpoly4 15401 |
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