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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 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 2 | bpolyval 16085 | . . 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 690 | . 2 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) |
| 4 | exp1 14108 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑1) = 𝑋) | |
| 5 | 1m1e0 12338 | . . . . . 6 ⊢ (1 − 1) = 0 | |
| 6 | 5 | oveq2i 7442 | . . . . 5 ⊢ (0...(1 − 1)) = (0...0) |
| 7 | 6 | sumeq1i 15733 | . . . 4 ⊢ Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) |
| 8 | 0z 12624 | . . . . . 6 ⊢ 0 ∈ ℤ | |
| 9 | bpoly0 16086 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1) | |
| 10 | 9 | oveq1d 7446 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 2) = (1 / 2)) |
| 11 | 10 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 · (1 / 2))) |
| 12 | halfcn 12481 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ | |
| 13 | 12 | mullidi 11266 | . . . . . . . 8 ⊢ (1 · (1 / 2)) = (1 / 2) |
| 14 | 11, 13 | eqtrdi 2793 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 / 2)) |
| 15 | 14, 12 | eqeltrdi 2849 | . . . . . 6 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) |
| 16 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (1C𝑘) = (1C0)) | |
| 17 | bcn0 14349 | . . . . . . . . . 10 ⊢ (1 ∈ ℕ0 → (1C0) = 1) | |
| 18 | 1, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ (1C0) = 1 |
| 19 | 16, 18 | eqtrdi 2793 | . . . . . . . 8 ⊢ (𝑘 = 0 → (1C𝑘) = 1) |
| 20 | oveq1 7438 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) | |
| 21 | oveq2 7439 | . . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (1 − 𝑘) = (1 − 0)) | |
| 22 | 1m0e1 12387 | . . . . . . . . . . . 12 ⊢ (1 − 0) = 1 | |
| 23 | 21, 22 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ (𝑘 = 0 → (1 − 𝑘) = 1) |
| 24 | 23 | oveq1d 7446 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = (1 + 1)) |
| 25 | df-2 12329 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
| 26 | 24, 25 | eqtr4di 2795 | . . . . . . . . 9 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = 2) |
| 27 | 20, 26 | oveq12d 7449 | . . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 2)) |
| 28 | 19, 27 | oveq12d 7449 | . . . . . . 7 ⊢ (𝑘 = 0 → ((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
| 29 | 28 | fsum1 15783 | . . . . . 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 2777 | . . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
| 32 | 7, 31 | eqtrid 2789 | . . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
| 33 | 4, 32 | oveq12d 7449 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)))) = (𝑋 − (1 / 2))) |
| 34 | 3, 33 | eqtrd 2777 | 1 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 / cdiv 11920 2c2 12321 ℕ0cn0 12526 ℤcz 12613 ...cfz 13547 ↑cexp 14102 Ccbc 14341 Σcsu 15722 BernPoly cbp 16082 |
| 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-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 |
| 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-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 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-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 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-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-bpoly 16083 |
| This theorem is referenced by: bpoly2 16093 bpoly3 16094 bpoly4 16095 |
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