Theorem bpolycl 16008

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 7367	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
2	1	eleq1d 2822	. . . 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 7367	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
5	4	eleq1d 2822	. . . 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 3163	. . . 4 ⊢ (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ))
8		bpolyval 16005	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋↑𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛 − 𝑚) + 1)))))
9	8	3adant3 1133	. . . . . . 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 1138	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑋 ∈ ℂ)
11		simp1 1137	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℕ0)
12	10, 11	expcld 14099	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋↑𝑛) ∈ ℂ)
13		fzfid 13926	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ∈ Fin)
14		elfzelz 13469	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...(𝑛 − 1)) → 𝑚 ∈ ℤ)
15		bccl 14275	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑚 ∈ ℤ) → (𝑛C𝑚) ∈ ℕ0)
16	11, 14, 15	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℕ0)
17	16	nn0cnd 12491	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℂ)
18		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
19	18	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((𝑘 BernPoly 𝑋) ∈ ℂ ↔ (𝑚 BernPoly 𝑋) ∈ ℂ))
20	19	rspccva 3564	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
21	20	3ad2antl3 1189	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
22		fzssp1 13512	. . . . . . . . . . . . . . 15 ⊢ (0...(𝑛 − 1)) ⊆ (0...((𝑛 − 1) + 1))
23	11	nn0cnd 12491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℂ)
24		ax-1cn 11087	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
25		npcan 11393	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
26	23, 24, 25	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
27	26	oveq2d 7376	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...((𝑛 − 1) + 1)) = (0...𝑛))
28	22, 27	sseqtrid 3965	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ⊆ (0...𝑛))
29	28	sselda 3922	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → 𝑚 ∈ (0...𝑛))
30		fznn0sub 13501	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑛) → (𝑛 − 𝑚) ∈ ℕ0)
31		nn0p1nn 12467	. . . . . . . . . . . . 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 12181	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛 − 𝑚) + 1) ∈ ℂ)
34	32	nnne0d 12218	. . . . . . . . . . 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 11922	. . . . . . . . . 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 11156	. . . . . . . . 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 15686	. . . . . . . 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 11496	. . . . . . 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 2837	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) ∈ ℂ)
40	39	3exp 1120	. . . . 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 242	. . 3 ⊢ (𝑛 ∈ ℕ0 → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
43	3, 6, 42	nn0sinds 13942	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ))
44	43	imp 406	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  (class class class)co 7360  ℂcc 11027  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   − cmin 11368   / cdiv 11798  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ...cfz 13452  ↑cexp 14014  Ccbc 14255  Σcsu 15639   BernPoly cbp 16002

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-fac 14227  df-bc 14256  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640  df-bpoly 16003

This theorem is referenced by:  bpolysum  16009  bpolydiflem  16010  fsumkthpow  16012  bpoly3  16014  bpoly4  16015