Theorem bpolycl 16250

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 7453	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
2	1	eleq1d 2849	. . . 4 ⊢ (𝑛 = 𝑘 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑘 BernPoly 𝑋) ∈ ℂ))
3	2	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑘 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ)))
4		oveq1 7453	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
5	4	eleq1d 2849	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑁 BernPoly 𝑋) ∈ ℂ))
6	5	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑁 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ)))
7		r19.21v 3191	. . . 4 ⊢ (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ))
8		bpolyval 16247	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))))
9	8	3adant3 1150	. . . . . . 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 1155	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑋 ∈ ℂ)
11		simp1 1154	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℕ0)
12	10, 11	expcld 14319	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋↑𝑛) ∈ ℂ)
13		fzfid 14146	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ∈ Fin)
14		elfzelz 13686	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝑛 − 1)) → 𝑚 ∈ ℤ)
15		bccl 14496	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℤ) → (𝑛C𝑚) ∈ ℕ0)
16	11, 14, 15	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℕ0)
17	16	nn0cnd 12699	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℂ)
18		oveq1 7453	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
19	18	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((𝑘 BernPoly 𝑋) ∈ ℂ ↔ (𝑚 BernPoly 𝑋) ∈ ℂ))
20	19	rspccva 3579	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
21	20	3ad2antl3 1206	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
22		fzssp1 13731	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑛 − 1)) ⊆ (0...((𝑛 − 1) + 1))
23	11	nn0cnd 12699	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℂ)
24		ax-1cn 11288	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
25		npcan 11596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
26	23, 24, 25	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
27	26	oveq2d 7462	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...((𝑛 − 1) + 1)) = (0...𝑛))
28	22, 27	sseqtrid 3955	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ⊆ (0...𝑛))
29	28	sselda 3913	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → 𝑚 ∈ (0...𝑛))
30		fznn0sub 13720	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑛) → (𝑛 − 𝑚) ∈ ℕ0)
31		nn0p1nn 12675	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 𝑚) ∈ ℕ0 → ((𝑛 − 𝑚) + 1) ∈ ℕ)
32	29, 30, 31	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℕ)
33	32	nncnd 12381	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℂ)
34	32	nnne0d 12418	. . . . . . . . . . 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 12123	. . . . . . . . . 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 11359	. . . . . . . . 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 15931	. . . . . . . 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 11699	. . . . . . 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 2864	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) ∈ ℂ)
40	39	3exp 1137	. . . . 5 ⊢ (𝑛 ∈ ℕ0 → (𝑋 ∈ ℂ → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
41	40	a2d 30	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
42	7, 41	biimtrid 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 14162	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ))
44	43	imp 412	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  (class class class)co 7446  ℂcc 11228  0cc0 11230  1c1 11231   + caddc 11233   · cmul 11235   − cmin 11571   / cdiv 12003  ℕcn 12365  ℕ₀cn0 12636  ℤcz 12723  ...cfz 13669  ↑cexp 14234  Ccbc 14476  Σcsu 15885   BernPoly cbp 16244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-oi 9534  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-n0 12637  df-z 12724  df-uz 12996  df-rp 13151  df-fz 13670  df-fzo 13819  df-seq 14175  df-exp 14235  df-fac 14448  df-bc 14477  df-hash 14505  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-clim 15687  df-sum 15886  df-bpoly 16245

This theorem is used by:  bpolysum  16251  bpolydiflem  16252  fsumkthpow  16254  bpoly3  16256  bpoly4  16257