Theorem bpolydif 15401

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.

Step	Hyp	Ref	Expression
1		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly (𝑋 + 1)) = (𝑘 BernPoly (𝑋 + 1)))
2		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 BernPoly 𝑋) = (𝑘 BernPoly 𝑋))
3	1, 2	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)))
4		id 22	. . . . . 6 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
5		oveq1 7142	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 − 1) = (𝑘 − 1))
6	5	oveq2d 7151	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑋↑(𝑛 − 1)) = (𝑋↑(𝑘 − 1)))
7	4, 6	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 · (𝑋↑(𝑛 − 1))) = (𝑘 · (𝑋↑(𝑘 − 1))))
8	3, 7	eqeq12d 2814	. . . 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)))))
9	8	imbi2d 344	. . 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 7142	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly (𝑋 + 1)) = (𝑁 BernPoly (𝑋 + 1)))
11		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 BernPoly 𝑋) = (𝑁 BernPoly 𝑋))
12	10, 11	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑛 BernPoly (𝑋 + 1)) − (𝑛 BernPoly 𝑋)) = ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)))
13		id 22	. . . . . 6 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
14		oveq1 7142	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
15	14	oveq2d 7151	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑋↑(𝑛 − 1)) = (𝑋↑(𝑁 − 1)))
16	13, 15	oveq12d 7153	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 · (𝑋↑(𝑛 − 1))) = (𝑁 · (𝑋↑(𝑁 − 1))))
17	12, 16	eqeq12d 2814	. . . 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)))))
18	17	imbi2d 344	. . 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 1133	. . . . 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 1135	. . . . 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 1190	. . . . . 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 7142	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly (𝑋 + 1)) = (𝑚 BernPoly (𝑋 + 1)))
23		oveq1 7142	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑘 BernPoly 𝑋) = (𝑚 BernPoly 𝑋))
24	22, 23	oveq12d 7153	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = ((𝑚 BernPoly (𝑋 + 1)) − (𝑚 BernPoly 𝑋)))
25		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → 𝑘 = 𝑚)
26		oveq1 7142	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑘 − 1) = (𝑚 − 1))
27	26	oveq2d 7151	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑋↑(𝑘 − 1)) = (𝑋↑(𝑚 − 1)))
28	25, 27	oveq12d 7153	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑘 · (𝑋↑(𝑘 − 1))) = (𝑚 · (𝑋↑(𝑚 − 1))))
29	24, 28	eqeq12d 2814	. . . . . . . . 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)))))
30	29	imbi2d 344	. . . . . . . 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))))))
31	30	rspccva 3570	. . . . . . 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)))))
32	31	3ad2antl2 1183	. . . . . 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)))))
33	21, 32	mpd 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))))
34	19, 20, 33	bpolydiflem 15400	. . . 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))))
35	34	3exp 1116	. . 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))))))
36	9, 18, 35	nnsinds 13351	. 2 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ℂ → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1)))))
37	36	imp 410	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ ℂ) → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  (class class class)co 7135  ℂcc 10524  1c1 10527   + caddc 10529   · cmul 10531   − cmin 10859  ℕcn 11625  ...cfz 12885  ↑cexp 13425   BernPoly cbp 15392

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-bpoly 15393

This theorem is referenced by:  fsumkthpow  15402