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Theorem bpolycl 14703
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 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)

Proof of Theorem bpolycl
Dummy variables 𝑛 𝑘 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6612 . . . . 5 (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
21eleq1d 2688 . . . 4 (𝑛 = 𝑘 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑘 BernPoly 𝑋) ∈ ℂ))
32imbi2d 330 . . 3 (𝑛 = 𝑘 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ)))
4 oveq1 6612 . . . . 5 (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
54eleq1d 2688 . . . 4 (𝑛 = 𝑁 → ((𝑛 BernPoly 𝑋) ∈ ℂ ↔ (𝑁 BernPoly 𝑋) ∈ ℂ))
65imbi2d 330 . . 3 (𝑛 = 𝑁 → ((𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ)))
7 r19.21v 2959 . . . 4 (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) ↔ (𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ))
8 bpolyval 14700 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1)))))
983adant3 1079 . . . . . . 7 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) = ((𝑋𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1)))))
10 simp2 1060 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑋 ∈ ℂ)
11 simp1 1059 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℕ0)
1210, 11expcld 12945 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋𝑛) ∈ ℂ)
13 fzfid 12709 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ∈ Fin)
14 elfzelz 12281 . . . . . . . . . . . 12 (𝑚 ∈ (0...(𝑛 − 1)) → 𝑚 ∈ ℤ)
15 bccl 13046 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0𝑚 ∈ ℤ) → (𝑛C𝑚) ∈ ℕ0)
1611, 14, 15syl2an 494 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℕ0)
1716nn0cnd 11298 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑛C𝑚) ∈ ℂ)
18 oveq1 6612 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
1918eleq1d 2688 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((𝑘 BernPoly 𝑋) ∈ ℂ ↔ (𝑚 BernPoly 𝑋) ∈ ℂ))
2019rspccva 3299 . . . . . . . . . . . 12 ((∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
21203ad2antl3 1223 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → (𝑚 BernPoly 𝑋) ∈ ℂ)
22 fzssp1 12323 . . . . . . . . . . . . . . 15 (0...(𝑛 − 1)) ⊆ (0...((𝑛 − 1) + 1))
2311nn0cnd 11298 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → 𝑛 ∈ ℂ)
24 ax-1cn 9939 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
25 npcan 10235 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
2623, 24, 25sylancl 693 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
2726oveq2d 6621 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...((𝑛 − 1) + 1)) = (0...𝑛))
2822, 27syl5sseq 3637 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (0...(𝑛 − 1)) ⊆ (0...𝑛))
2928sselda 3588 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → 𝑚 ∈ (0...𝑛))
30 fznn0sub 12312 . . . . . . . . . . . . 13 (𝑚 ∈ (0...𝑛) → (𝑛𝑚) ∈ ℕ0)
31 nn0p1nn 11277 . . . . . . . . . . . . 13 ((𝑛𝑚) ∈ ℕ0 → ((𝑛𝑚) + 1) ∈ ℕ)
3229, 30, 313syl 18 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛𝑚) + 1) ∈ ℕ)
3332nncnd 10981 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛𝑚) + 1) ∈ ℂ)
3432nnne0d 11010 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛𝑚) + 1) ≠ 0)
3521, 33, 34divcld 10746 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1)) ∈ ℂ)
3617, 35mulcld 10005 . . . . . . . . 9 (((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) ∧ 𝑚 ∈ (0...(𝑛 − 1))) → ((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1))) ∈ ℂ)
3713, 36fsumcl 14392 . . . . . . . 8 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1))) ∈ ℂ)
3812, 37subcld 10337 . . . . . . 7 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → ((𝑋𝑛) − Σ𝑚 ∈ (0...(𝑛 − 1))((𝑛C𝑚) · ((𝑚 BernPoly 𝑋) / ((𝑛𝑚) + 1)))) ∈ ℂ)
399, 38eqeltrd 2704 . . . . . 6 ((𝑛 ∈ ℕ0𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑛 BernPoly 𝑋) ∈ ℂ)
40393exp 1261 . . . . 5 (𝑛 ∈ ℕ0 → (𝑋 ∈ ℂ → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
4140a2d 29 . . . 4 (𝑛 ∈ ℕ0 → ((𝑋 ∈ ℂ → ∀𝑘 ∈ (0...(𝑛 − 1))(𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
427, 41syl5bi 232 . . 3 (𝑛 ∈ ℕ0 → (∀𝑘 ∈ (0...(𝑛 − 1))(𝑋 ∈ ℂ → (𝑘 BernPoly 𝑋) ∈ ℂ) → (𝑋 ∈ ℂ → (𝑛 BernPoly 𝑋) ∈ ℂ)))
433, 6, 42nn0sinds 12725 . 2 (𝑁 ∈ ℕ0 → (𝑋 ∈ ℂ → (𝑁 BernPoly 𝑋) ∈ ℂ))
4443imp 445 1 ((𝑁 ∈ ℕ0𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  (class class class)co 6605  cc 9879  0cc0 9881  1c1 9882   + caddc 9884   · cmul 9886  cmin 10211   / cdiv 10629  cn 10965  0cn0 11237  cz 11322  ...cfz 12265  cexp 12797  Ccbc 13026  Σcsu 14345   BernPoly cbp 14697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-fac 12998  df-bc 13027  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346  df-bpoly 14698
This theorem is referenced by:  bpolysum  14704  bpolydiflem  14705  fsumkthpow  14707  bpoly3  14709  bpoly4  14710
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