Theorem bpolycl 15946

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 7369	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
2	1	eleq1d 2817	. . . 4 ⊢ (𝑛 = 𝑘 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑘 BernPoly 𝑋) ∈ ℂ))
3	2	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑘 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ)))
4		oveq1 7369	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
5	4	eleq1d 2817	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑁 BernPoly 𝑋) ∈ ℂ))
6	5	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ)))
7		r19.21v 3172	. . . 4 ⊢ (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ))
8		bpolyval 15943	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))))
9	8	3adant3 1132	. . . . . . 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 1137	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑋 ∈ ℂ)
11		simp1 1136	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℕ0)
12	10, 11	expcld 14061	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋↑𝑛) ∈ ℂ)
13		fzfid 13888	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ∈ Fin)
14		elfzelz 13451	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝑛 − 1)) → 𝑚 ∈ ℤ)
15		bccl 14232	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℤ) → (𝑛C𝑚) ∈ ℕ0)
16	11, 14, 15	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℕ0)
17	16	nn0cnd 12484	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℂ)
18		oveq1 7369	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
19	18	eleq1d 2817	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((𝑘 BernPoly 𝑋) ∈ ℂ ↔ (𝑚 BernPoly 𝑋) ∈ ℂ))
20	19	rspccva 3581	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
21	20	3ad2antl3 1187	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
22		fzssp1 13494	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑛 − 1)) ⊆ (0...((𝑛 − 1) + 1))
23	11	nn0cnd 12484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℂ)
24		ax-1cn 11118	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
25		npcan 11419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
26	23, 24, 25	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
27	26	oveq2d 7378	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...((𝑛 − 1) + 1)) = (0...𝑛))
28	22, 27	sseqtrid 3999	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ⊆ (0...𝑛))
29	28	sselda 3947	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → 𝑚 ∈ (0...𝑛))
30		fznn0sub 13483	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑛) → (𝑛 − 𝑚) ∈ ℕ0)
31		nn0p1nn 12461	. . . . . . . . . . . . 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 12178	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℂ)
34	32	nnne0d 12212	. . . . . . . . . . 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 11940	. . . . . . . . . 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 11184	. . . . . . . . 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 15629	. . . . . . . 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 11521	. . . . . . 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 2832	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) ∈ ℂ)
40	39	3exp 1119	. . . . 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	biimtrid 241	. . 3 ⊢ (𝑛 ∈ ℕ0 → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
43	3, 6, 42	nn0sinds 13904	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ))
44	43	imp 407	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  (class class class)co 7362  ℂcc 11058  0cc0 11060  1c1 11061   + caddc 11063   · cmul 11065   − cmin 11394   / cdiv 11821  ℕcn 12162  ℕ₀cn0 12422  ℤcz 12508  ...cfz 13434  ↑cexp 13977  Ccbc 14212  Σcsu 15582   BernPoly cbp 15940

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-oi 9455  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-fz 13435  df-fzo 13578  df-seq 13917  df-exp 13978  df-fac 14184  df-bc 14213  df-hash 14241  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-clim 15382  df-sum 15583  df-bpoly 15941

This theorem is referenced by:  bpolysum  15947  bpolydiflem  15948  fsumkthpow  15950  bpoly3  15952  bpoly4  15953