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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 11185 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | bpolyval 14619 | . . 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 702 | . 2 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))))) |
4 | exp1 12728 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑1) = 𝑋) | |
5 | 1m1e0 10966 | . . . . . 6 ⊢ (1 − 1) = 0 | |
6 | 5 | oveq2i 6560 | . . . . 5 ⊢ (0...(1 − 1)) = (0...0) |
7 | 6 | sumeq1i 14276 | . . . 4 ⊢ Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) |
8 | 0z 11265 | . . . . . 6 ⊢ 0 ∈ ℤ | |
9 | bpoly0 14620 | . . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1) | |
10 | 9 | oveq1d 6564 | . . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 2) = (1 / 2)) |
11 | 10 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 · (1 / 2))) |
12 | halfcn 11124 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ | |
13 | 12 | mulid2i 9922 | . . . . . . . 8 ⊢ (1 · (1 / 2)) = (1 / 2) |
14 | 11, 13 | syl6eq 2660 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) = (1 / 2)) |
15 | 14, 12 | syl6eqel 2696 | . . . . . 6 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 2)) ∈ ℂ) |
16 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (1C𝑘) = (1C0)) | |
17 | bcn0 12959 | . . . . . . . . . 10 ⊢ (1 ∈ ℕ0 → (1C0) = 1) | |
18 | 1, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ (1C0) = 1 |
19 | 16, 18 | syl6eq 2660 | . . . . . . . 8 ⊢ (𝑘 = 0 → (1C𝑘) = 1) |
20 | oveq1 6556 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋)) | |
21 | oveq2 6557 | . . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (1 − 𝑘) = (1 − 0)) | |
22 | 1m0e1 11008 | . . . . . . . . . . . 12 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2660 | . . . . . . . . . . 11 ⊢ (𝑘 = 0 → (1 − 𝑘) = 1) |
24 | 23 | oveq1d 6564 | . . . . . . . . . 10 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = (1 + 1)) |
25 | df-2 10956 | . . . . . . . . . 10 ⊢ 2 = (1 + 1) | |
26 | 24, 25 | syl6eqr 2662 | . . . . . . . . 9 ⊢ (𝑘 = 0 → ((1 − 𝑘) + 1) = 2) |
27 | 20, 26 | oveq12d 6567 | . . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 2)) |
28 | 19, 27 | oveq12d 6567 | . . . . . . 7 ⊢ (𝑘 = 0 → ((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
29 | 28 | fsum1 14320 | . . . . . 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 694 | . . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 2))) |
31 | 30, 14 | eqtrd 2644 | . . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
32 | 7, 31 | syl5eq 2656 | . . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1))) = (1 / 2)) |
33 | 4, 32 | oveq12d 6567 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑1) − Σ𝑘 ∈ (0...(1 − 1))((1C𝑘) · ((𝑘 BernPoly 𝑋) / ((1 − 𝑘) + 1)))) = (𝑋 − (1 / 2))) |
34 | 3, 33 | eqtrd 2644 | 1 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 / cdiv 10563 2c2 10947 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 ↑cexp 12722 Ccbc 12951 Σcsu 14264 BernPoly cbp 14616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-bpoly 14617 |
This theorem is referenced by: bpoly2 14627 bpoly3 14628 bpoly4 14629 |
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