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Theorem bpolydif 14711
Description: Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
bpolydif ((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Proof of Theorem bpolydif
Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6611 . . . . . 6 (𝑛 = 𝑘 → (𝑛 BernPoly (𝑋 + 1)) = (𝑘 BernPoly (𝑋 + 1)))
2 oveq1 6611 . . . . . 6 (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
31, 2oveq12d 6622 . . . . 5 (𝑛 = 𝑘 → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)))
4 id 22 . . . . . 6 (𝑛 = 𝑘𝑛 = 𝑘)
5 oveq1 6611 . . . . . . 7 (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
65oveq2d 6620 . . . . . 6 (𝑛 = 𝑘 → (𝑋↑(𝑛 − 1)) = (𝑋↑(𝑘 − 1)))
74, 6oveq12d 6622 . . . . 5 (𝑛 = 𝑘 → (𝑛 · (𝑋↑(𝑛 − 1))) = (𝑘 · (𝑋↑(𝑘 − 1))))
83, 7eqeq12d 2636 . . . 4 (𝑛 = 𝑘 → (((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1))) ↔ ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))))
98imbi2d 330 . . 3 (𝑛 = 𝑘 → ((𝑋 ∈ ℂ → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1)))) ↔ (𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))))))
10 oveq1 6611 . . . . . 6 (𝑛 = 𝑁 → (𝑛 BernPoly (𝑋 + 1)) = (𝑁 BernPoly (𝑋 + 1)))
11 oveq1 6611 . . . . . 6 (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
1210, 11oveq12d 6622 . . . . 5 (𝑛 = 𝑁 → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)))
13 id 22 . . . . . 6 (𝑛 = 𝑁𝑛 = 𝑁)
14 oveq1 6611 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
1514oveq2d 6620 . . . . . 6 (𝑛 = 𝑁 → (𝑋↑(𝑛 − 1)) = (𝑋↑(𝑁 − 1)))
1613, 15oveq12d 6622 . . . . 5 (𝑛 = 𝑁 → (𝑛 · (𝑋↑(𝑛 − 1))) = (𝑁 · (𝑋↑(𝑁 − 1))))
1712, 16eqeq12d 2636 . . . 4 (𝑛 = 𝑁 → (((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1))) ↔ ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1)))))
1817imbi2d 330 . . 3 (𝑛 = 𝑁 → ((𝑋 ∈ ℂ → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1)))) ↔ (𝑋 ∈ ℂ → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))))
19 simp1 1059 . . . . 5 ((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) → 𝑛 ∈ ℕ)
20 simp3 1061 . . . . 5 ((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) → 𝑋 ∈ ℂ)
21 simpl3 1064 . . . . . 6 (((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ (1...(𝑛 − 1))) → 𝑋 ∈ ℂ)
22 oveq1 6611 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑘 BernPoly (𝑋 + 1)) = (𝑚 BernPoly (𝑋 + 1)))
23 oveq1 6611 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
2422, 23oveq12d 6622 . . . . . . . . . 10 (𝑘 = 𝑚 → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)))
25 id 22 . . . . . . . . . . 11 (𝑘 = 𝑚𝑘 = 𝑚)
26 oveq1 6611 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (𝑘 − 1) = (𝑚 − 1))
2726oveq2d 6620 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝑋↑(𝑘 − 1)) = (𝑋↑(𝑚 − 1)))
2825, 27oveq12d 6622 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝑘 · (𝑋↑(𝑘 − 1))) = (𝑚 · (𝑋↑(𝑚 − 1))))
2924, 28eqeq12d 2636 . . . . . . . . 9 (𝑘 = 𝑚 → (((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))) ↔ ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)) = (𝑚 · (𝑋↑(𝑚 − 1)))))
3029imbi2d 330 . . . . . . . 8 (𝑘 = 𝑚 → ((𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ↔ (𝑋 ∈ ℂ → ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)) = (𝑚 · (𝑋↑(𝑚 − 1))))))
3130rspccva 3294 . . . . . . 7 ((∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑚 ∈ (1...(𝑛 − 1))) → (𝑋 ∈ ℂ → ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)) = (𝑚 · (𝑋↑(𝑚 − 1)))))
32313ad2antl2 1222 . . . . . 6 (((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ (1...(𝑛 − 1))) → (𝑋 ∈ ℂ → ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)) = (𝑚 · (𝑋↑(𝑚 − 1)))))
3321, 32mpd 15 . . . . 5 (((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ (1...(𝑛 − 1))) → ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)) = (𝑚 · (𝑋↑(𝑚 − 1))))
3419, 20, 33bpolydiflem 14710 . . . 4 ((𝑛 ∈ ℕ ∧ ∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) ∧ 𝑋 ∈ ℂ) → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1))))
35343exp 1261 . . 3 (𝑛 ∈ ℕ → (∀𝑘 ∈ (1...(𝑛 − 1))(𝑋 ∈ ℂ → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1)))) → (𝑋 ∈ ℂ → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = (𝑛 · (𝑋↑(𝑛 − 1))))))
369, 18, 35nnsinds 12727 . 2 (𝑁 ∈ ℕ → (𝑋 ∈ ℂ → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1)))))
3736imp 445 1 ((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  (class class class)co 6604  cc 9878  1c1 9881   + caddc 9883   · cmul 9885  cmin 10210  cn 10964  ...cfz 12268  cexp 12800   BernPoly cbp 14702
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-bpoly 14703
This theorem is referenced by:  fsumkthpow  14712
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