bpolysum - Metamath Proof Explorer

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Theorem bpolysum 16039

Description: A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)

**Assertion**
Ref	Expression
bpolysum	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (𝑋↑𝑁))

Distinct variable groups: 𝑘,𝑁 𝑘,𝑋

**Proof of Theorem bpolysum**
Step	Hyp	Ref	Expression
1		simpl 481	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → 𝑁 ∈ ℕ₀)
2		nn0uz 12904	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2839	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → 𝑁 ∈ (ℤ_≥‘0))
4		elfzelz 13543	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
5		bccl 14323	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
6	1, 4, 5	syl2an 594	. . . . 5 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ₀)
7	6	nn0cnd 12574	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ)
8		elfznn0 13636	. . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
9		simpr 483	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → 𝑋 ∈ ℂ)
10		bpolycl 16038	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑘 BernPoly 𝑋) ∈ ℂ)
11	8, 9, 10	syl2anr 595	. . . . 5 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 BernPoly 𝑋) ∈ ℂ)
12		fznn0sub 13575	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ ℕ₀)
13	12	adantl 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℕ₀)
14		nn0p1nn 12551	. . . . . . 7 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ → ((𝑁 − 𝑘) + 1) ∈ ℕ)
15	13, 14	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ∈ ℕ)
16	15	nncnd 12268	. . . . 5 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ∈ ℂ)
17	15	nnne0d 12302	. . . . 5 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 1) ≠ 0)
18	11, 16, 17	divcld 12030	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
19	7, 18	mulcld 11274	. . 3 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
20		oveq2 7434	. . . 4 ⊢ (𝑘 = 𝑁 → (𝑁C𝑘) = (𝑁C𝑁))
21		oveq1 7433	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑘 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
22		oveq2 7434	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑁 − 𝑘) = (𝑁 − 𝑁))
23	22	oveq1d 7441	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑁 − 𝑘) + 1) = ((𝑁 − 𝑁) + 1))
24	21, 23	oveq12d 7444	. . . 4 ⊢ (𝑘 = 𝑁 → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)))
25	20, 24	oveq12d 7444	. . 3 ⊢ (𝑘 = 𝑁 → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))))
26	3, 19, 25	fsumm1 15739	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)))))
27		bcnn 14313	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑁C𝑁) = 1)
28	27	adantr 479	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁C𝑁) = 1)
29		nn0cn 12522	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
30	29	adantr 479	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → 𝑁 ∈ ℂ)
31	30	subidd 11599	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 − 𝑁) = 0)
32	31	oveq1d 7441	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 − 𝑁) + 1) = (0 + 1))
33		0p1e1 12374	. . . . . . . 8 ⊢ (0 + 1) = 1
34	32, 33	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 − 𝑁) + 1) = 1)
35	34	oveq2d 7442	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)) = ((𝑁 BernPoly 𝑋) / 1))
36		bpolycl 16038	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)
37	36	div1d 12022	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly 𝑋) / 1) = (𝑁 BernPoly 𝑋))
38	35, 37	eqtrd 2768	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)) = (𝑁 BernPoly 𝑋))
39	28, 38	oveq12d 7444	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))) = (1 · (𝑁 BernPoly 𝑋)))
40	36	mullidd 11272	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (1 · (𝑁 BernPoly 𝑋)) = (𝑁 BernPoly 𝑋))
41	39, 40	eqtrd 2768	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1))) = (𝑁 BernPoly 𝑋))
42	41	oveq2d 7442	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + ((𝑁C𝑁) · ((𝑁 BernPoly 𝑋) / ((𝑁 − 𝑁) + 1)))) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋)))
43		bpolyval 16035	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
44	43	eqcomd 2734	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (𝑁 BernPoly 𝑋))
45		expcl 14086	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝑋↑𝑁) ∈ ℂ)
46	45	ancoms 457	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑋↑𝑁) ∈ ℂ)
47		fzfid 13980	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (0...(𝑁 − 1)) ∈ Fin)
48		fzssp1 13586	. . . . . . . 8 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
49		ax-1cn 11206	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
50		npcan 11509	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
51	30, 49, 50	sylancl 584	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
52	51	oveq2d 7442	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (0...((𝑁 − 1) + 1)) = (0...𝑁))
53	48, 52	sseqtrid 4034	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
54	53	sselda 3982	. . . . . 6 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁))
55	54, 19	syldan 589	. . . . 5 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
56	47, 55	fsumcl 15721	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
57	46, 56, 36	subaddd 11629	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (𝑁 BernPoly 𝑋) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋)) = (𝑋↑𝑁)))
58	44, 57	mpbid 231	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) + (𝑁 BernPoly 𝑋)) = (𝑋↑𝑁))
59	26, 42, 58	3eqtrd 2772	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (𝑋↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 ℂcc 11146 0cc0 11148 1c1 11149 + caddc 11151 · cmul 11153 − cmin 11484 / cdiv 11911 ℕcn 12252 ℕ₀cn0 12512 ℤcz 12598 ℤ_≥cuz 12862 ...cfz 13526 ↑cexp 14068 Ccbc 14303 Σcsu 15674 BernPoly cbp 16032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-sum 15675 df-bpoly 16033

This theorem is referenced by: (None)

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