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Theorem bpolyval 16035

Description: The value of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.)

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

Distinct variable groups: 𝑘,𝑁 𝑘,𝑋

Proof of Theorem bpolyval Dummy variables 𝑔 𝑚 𝑛 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6915	. . . . . 6 ⊢ (♯‘dom 𝑐) ∈ V
2		oveq2 7434	. . . . . . 7 ⊢ (𝑛 = (♯‘dom 𝑐) → (𝑋↑𝑛) = (𝑋↑(♯‘dom 𝑐)))
3		oveq1 7433	. . . . . . . . 9 ⊢ (𝑛 = (♯‘dom 𝑐) → (𝑛C𝑚) = ((♯‘dom 𝑐)C𝑚))
4		oveq1 7433	. . . . . . . . . . 11 ⊢ (𝑛 = (♯‘dom 𝑐) → (𝑛 − 𝑚) = ((♯‘dom 𝑐) − 𝑚))
5	4	oveq1d 7441	. . . . . . . . . 10 ⊢ (𝑛 = (♯‘dom 𝑐) → ((𝑛 − 𝑚) + 1) = (((♯‘dom 𝑐) − 𝑚) + 1))
6	5	oveq2d 7442	. . . . . . . . 9 ⊢ (𝑛 = (♯‘dom 𝑐) → ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)) = ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1)))
7	3, 6	oveq12d 7444	. . . . . . . 8 ⊢ (𝑛 = (♯‘dom 𝑐) → ((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1))) = (((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1))))
8	7	sumeq2sdv 15692	. . . . . . 7 ⊢ (𝑛 = (♯‘dom 𝑐) → Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1))) = Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1))))
9	2, 8	oveq12d 7444	. . . . . 6 ⊢ (𝑛 = (♯‘dom 𝑐) → ((𝑋↑𝑛) − Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)))) = ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1)))))
10	1, 9	csbie 3930	. . . . 5 ⊢ ⦋(♯‘dom 𝑐) / 𝑛⦌((𝑋↑𝑛) − Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)))) = ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1))))
11		oveq2 7434	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (𝑛C𝑚) = (𝑛C𝑘))
12		fveq2 6902	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑐‘𝑚) = (𝑐‘𝑘))
13		oveq2 7434	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝑛 − 𝑚) = (𝑛 − 𝑘))
14	13	oveq1d 7441	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝑛 − 𝑚) + 1) = ((𝑛 − 𝑘) + 1))
15	12, 14	oveq12d 7444	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)) = ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1)))
16	11, 15	oveq12d 7444	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → ((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1))) = ((𝑛C𝑘) · ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1))))
17	16	cbvsumv 15684	. . . . . . . 8 ⊢ Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1))) = Σ𝑘 ∈ dom 𝑐((𝑛C𝑘) · ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1)))
18		dmeq 5910	. . . . . . . . 9 ⊢ (𝑐 = 𝑔 → dom 𝑐 = dom 𝑔)
19		fveq1 6901	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑔 → (𝑐‘𝑘) = (𝑔‘𝑘))
20	19	oveq1d 7441	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑔 → ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1)) = ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))
21	20	oveq2d 7442	. . . . . . . . . 10 ⊢ (𝑐 = 𝑔 → ((𝑛C𝑘) · ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1))) = ((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))
22	21	adantr 479	. . . . . . . . 9 ⊢ ((𝑐 = 𝑔 ∧ 𝑘 ∈ dom 𝑐) → ((𝑛C𝑘) · ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1))) = ((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))
23	18, 22	sumeq12dv 15694	. . . . . . . 8 ⊢ (𝑐 = 𝑔 → Σ𝑘 ∈ dom 𝑐((𝑛C𝑘) · ((𝑐‘𝑘) / ((𝑛 − 𝑘) + 1))) = Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))
24	17, 23	eqtrid 2780	. . . . . . 7 ⊢ (𝑐 = 𝑔 → Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1))) = Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))
25	24	oveq2d 7442	. . . . . 6 ⊢ (𝑐 = 𝑔 → ((𝑋↑𝑛) − Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)))) = ((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
26	25	csbeq2dv 3901	. . . . 5 ⊢ (𝑐 = 𝑔 → ⦋(♯‘dom 𝑐) / 𝑛⦌((𝑋↑𝑛) − Σ𝑚 ∈ dom 𝑐((𝑛C𝑚) · ((𝑐‘𝑚) / ((𝑛 − 𝑚) + 1)))) = ⦋(♯‘dom 𝑐) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
27	10, 26	eqtr3id 2782	. . . 4 ⊢ (𝑐 = 𝑔 → ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1)))) = ⦋(♯‘dom 𝑐) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
28	18	fveq2d 6906	. . . . 5 ⊢ (𝑐 = 𝑔 → (♯‘dom 𝑐) = (♯‘dom 𝑔))
29	28	csbeq1d 3898	. . . 4 ⊢ (𝑐 = 𝑔 → ⦋(♯‘dom 𝑐) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
30	27, 29	eqtrd 2768	. . 3 ⊢ (𝑐 = 𝑔 → ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1)))) = ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
31	30	cbvmptv 5265	. 2 ⊢ (𝑐 ∈ V ↦ ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1))))) = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
32		eqid 2728	. 2 ⊢ wrecs( < , ℕ₀, (𝑐 ∈ V ↦ ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1)))))) = wrecs( < , ℕ₀, (𝑐 ∈ V ↦ ((𝑋↑(♯‘dom 𝑐)) − Σ𝑚 ∈ dom 𝑐(((♯‘dom 𝑐)C𝑚) · ((𝑐‘𝑚) / (((♯‘dom 𝑐) − 𝑚) + 1))))))
33	31, 32	bpolylem 16034	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⦋csb 3894 ↦ cmpt 5235 dom cdm 5682 ‘cfv 6553 (class class class)co 7426 wrecscwrecs 8325 ℂcc 11146 0cc0 11148 1c1 11149 + caddc 11151 · cmul 11153 < clt 11288 − cmin 11484 / cdiv 11911 ℕ₀cn0 12512 ...cfz 13526 ↑cexp 14068 Ccbc 14303 ♯chash 14331 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

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-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-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-seq 14009 df-hash 14332 df-sum 15675 df-bpoly 16033

This theorem is referenced by: bpoly0 16036 bpoly1 16037 bpolycl 16038 bpolysum 16039 bpolydiflem 16040 bpoly2 16043 bpoly3 16044 bpoly4 16045

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