Theorem bpolycl 15398

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

Assertion

Ref	Expression
bpolycl	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Proof of Theorem bpolycl

Dummy variables 𝑛 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7142	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
2	1	eleq1d 2874	. . . 4 ⊢ (𝑛 = 𝑘 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑘 BernPoly 𝑋) ∈ ℂ))
3	2	imbi2d 344	. . 3 ⊢ (𝑛 = 𝑘 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ)))
4		oveq1 7142	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
5	4	eleq1d 2874	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑁 BernPoly 𝑋) ∈ ℂ))
6	5	imbi2d 344	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ)))
7		r19.21v 3142	. . . 4 ⊢ (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ))
8		bpolyval 15395	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))))
9	8	3adant3 1129	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))))
10		simp2 1134	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑋 ∈ ℂ)
11		simp1 1133	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℕ0)
12	10, 11	expcld 13506	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋↑𝑛) ∈ ℂ)
13		fzfid 13336	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ∈ Fin)
14		elfzelz 12902	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝑛 − 1)) → 𝑚 ∈ ℤ)
15		bccl 13678	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℤ) → (𝑛C𝑚) ∈ ℕ0)
16	11, 14, 15	syl2an 598	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℕ0)
17	16	nn0cnd 11945	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℂ)
18		oveq1 7142	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
19	18	eleq1d 2874	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((𝑘 BernPoly 𝑋) ∈ ℂ ↔ (𝑚 BernPoly 𝑋) ∈ ℂ))
20	19	rspccva 3570	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
21	20	3ad2antl3 1184	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
22		fzssp1 12945	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑛 − 1)) ⊆ (0...((𝑛 − 1) + 1))
23	11	nn0cnd 11945	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℂ)
24		ax-1cn 10584	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
25		npcan 10884	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
26	23, 24, 25	sylancl 589	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
27	26	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...((𝑛 − 1) + 1)) = (0...𝑛))
28	22, 27	sseqtrid 3967	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ⊆ (0...𝑛))
29	28	sselda 3915	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → 𝑚 ∈ (0...𝑛))
30		fznn0sub 12934	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑛) → (𝑛 − 𝑚) ∈ ℕ0)
31		nn0p1nn 11924	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 𝑚) ∈ ℕ0 → ((𝑛 − 𝑚) + 1) ∈ ℕ)
32	29, 30, 31	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℕ)
33	32	nncnd 11641	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℂ)
34	32	nnne0d 11675	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ≠ 0)
35	21, 33, 34	divcld 11405	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)) ∈ ℂ)
36	17, 35	mulcld 10650	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1))) ∈ ℂ)
37	13, 36	fsumcl 15082	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1))) ∈ ℂ)
38	12, 37	subcld 10986	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))) ∈ ℂ)
39	9, 38	eqeltrd 2890	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) ∈ ℂ)
40	39	3exp 1116	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑋 ∈ ℂ → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
41	40	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
42	7, 41	syl5bi 245	. . 3 ⊢ (𝑛 ∈ ℕ0 → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
43	3, 6, 42	nn0sinds 13352	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ))
44	43	imp 410	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  (class class class)co 7135  ℂcc 10524  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   − cmin 10859   / cdiv 11286  ℕcn 11625  ℕ₀cn0 11885  ℤcz 11969  ...cfz 12885  ↑cexp 13425  Ccbc 13658  Σcsu 15034   BernPoly cbp 15392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-bpoly 15393

This theorem is referenced by:  bpolysum  15399  bpolydiflem  15400  fsumkthpow  15402  bpoly3  15404  bpoly4  15405