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Theorem bpolydiflem 16146

Description: Lemma for bpolydif 16147. (Contributed by Scott Fenton, 12-Jun-2014.)

**Hypotheses**
Ref	Expression
bpolydiflem.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
bpolydiflem.2	⊢ (𝜑 → 𝑋 ∈ ℂ)
bpolydiflem.3	⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))))

**Assertion**
Ref	Expression
bpolydiflem	⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Distinct variable groups: 𝑘,𝑁 𝜑,𝑘 𝑘,𝑋

Proof of Theorem bpolydiflem Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bpolydiflem.1	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnnn0d 12593	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		bpolydiflem.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ)
4		peano2cn 11410	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈ ℂ)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
6		bpolyval 16141	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑋 + 1) ∈ ℂ) → (𝑁 BernPoly (𝑋 + 1)) = (((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
7	2, 5, 6	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑁 BernPoly (𝑋 + 1)) = (((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
8		bpolyval 16141	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
9	2, 3, 8	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
10	7, 9	oveq12d 7435	. 2 ⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = ((((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))))
11	5, 2	expcld 14214	. . 3 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) ∈ ℂ)
12		fzfid 14041	. . . 4 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
13		elfzelz 13582	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
14		bccl 14390	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
15	2, 13, 14	syl2an 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℕ₀)
16	15	nn0cnd 12595	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ)
17		elfznn0 13679	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
18		bpolycl 16144	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑋 + 1) ∈ ℂ) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
19	17, 5, 18	syl2anr 609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
20		fzssp1 13626	. . . . . . . . . . 11 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
21	1	nncnd 12277	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
22		ax-1cn 11186	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
23		npcan 11494	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
24	21, 22, 23	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7433	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁))
26	20, 25	sseqtrid 3976	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
27	26	sselda 3934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁))
28		fznn0sub 13615	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ ℕ₀)
29	27, 28	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 𝑘) ∈ ℕ₀)
30		nn0p1nn 12571	. . . . . . . 8 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ → ((𝑁 − 𝑘) + 1) ∈ ℕ)
31	29, 30	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℕ)
32	31	nncnd 12277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ)
33	31	nnne0d 12314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0)
34	19, 32, 33	divcld 12019	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
35	16, 34	mulcld 11257	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
36	12, 35	fsumcl 15823	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
37	3, 2	expcld 14214	. . 3 ⊢ (𝜑 → (𝑋↑𝑁) ∈ ℂ)
38		bpolycl 16144	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑘 BernPoly 𝑋) ∈ ℂ)
39	17, 3, 38	syl2anr 609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ)
40	39, 32, 33	divcld 12019	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
41	16, 40	mulcld 11257	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
42	12, 41	fsumcl 15823	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
43	11, 36, 37, 42	sub4d 11646	. 2 ⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))))
44	26	sselda 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...𝑁))
45		bccl2 14391	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ)
46	45	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ)
47	46	nncnd 12277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ)
48		elfznn0 13679	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ₀)
49		expcl 14147	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝑋↑𝑚) ∈ ℂ)
50	3, 48, 49	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑋↑𝑚) ∈ ℂ)
51	47, 50	mulcld 11257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
52	44, 51	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
53	12, 52	fsumcl 15823	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
54		addcom 11424	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑋 + 1) = (1 + 𝑋))
55	3, 22, 54	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 + 1) = (1 + 𝑋))
56	55	oveq1d 7432	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = ((1 + 𝑋)↑𝑁))
57		binom1p 15924	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
58	3, 2, 57	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
59	56, 58	eqtrd 2797	. . . . . . 7 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
60		nn0uz 12929	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
61	2, 60	eleqtrdi 2872	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
62		oveq2 7425	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑁C𝑚) = (𝑁C𝑁))
63		oveq2 7425	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑋↑𝑚) = (𝑋↑𝑁))
64	62, 63	oveq12d 7435	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C𝑁) · (𝑋↑𝑁)))
65	61, 51, 64	fsumm1 15841	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C𝑁) · (𝑋↑𝑁))))
66		bcnn 14380	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁C𝑁) = 1)
67	2, 66	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁C𝑁) = 1)
68	67	oveq1d 7432	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (1 · (𝑋↑𝑁)))
69	37	mullidd 11255	. . . . . . . . 9 ⊢ (𝜑 → (1 · (𝑋↑𝑁)) = (𝑋↑𝑁))
70	68, 69	eqtrd 2797	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (𝑋↑𝑁))
71	70	oveq2d 7433	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C𝑁) · (𝑋↑𝑁))) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑋↑𝑁)))
72	59, 65, 71	3eqtrd 2801	. . . . . 6 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑋↑𝑁)))
73	53, 37, 72	mvrraddd 11654	. . . . 5 ⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)))
74		nnm1nn0 12573	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
75	1, 74	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
76	75, 60	eleqtrdi 2872	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘0))
77		oveq2 7425	. . . . . . 7 ⊢ (𝑚 = (𝑁 − 1) → (𝑁C𝑚) = (𝑁C(𝑁 − 1)))
78		oveq2 7425	. . . . . . 7 ⊢ (𝑚 = (𝑁 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑁 − 1)))
79	77, 78	oveq12d 7435	. . . . . 6 ⊢ (𝑚 = (𝑁 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))))
80	76, 52, 79	fsumm1 15841	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) = (Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1)))))
81		1cnd 11230	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
82	21, 81, 81	subsub4d 11628	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
83		df-2 12331	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
84	83	oveq2i 7428	. . . . . . . . 9 ⊢ (𝑁 − 2) = (𝑁 − (1 + 1))
85	82, 84	eqtr4di 2815	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
86	85	oveq2d 7433	. . . . . . 7 ⊢ (𝜑 → (0...((𝑁 − 1) − 1)) = (0...(𝑁 − 2)))
87	86	sumeq1d 15791	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)))
88		bcnm1 14395	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁C(𝑁 − 1)) = 𝑁)
89	2, 88	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑁C(𝑁 − 1)) = 𝑁)
90	89	oveq1d 7432	. . . . . 6 ⊢ (𝜑 → ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))) = (𝑁 · (𝑋↑(𝑁 − 1))))
91	87, 90	oveq12d 7435	. . . . 5 ⊢ (𝜑 → (Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1)))) = (Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))))
92	73, 80, 91	3eqtrd 2801	. . . 4 ⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = (Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))))
93		oveq2 7425	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑁C𝑘) = (𝑁C0))
94		oveq1 7424	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑘 BernPoly (𝑋 + 1)) = (0 BernPoly (𝑋 + 1)))
95		oveq2 7425	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑁 − 𝑘) = (𝑁 − 0))
96	95	oveq1d 7432	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((𝑁 − 𝑘) + 1) = ((𝑁 − 0) + 1))
97	94, 96	oveq12d 7435	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)))
98	93, 97	oveq12d 7435	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))))
99	76, 35, 98	fsum1p 15843	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
100		bpoly0 16142	. . . . . . . . . . 11 ⊢ ((𝑋 + 1) ∈ ℂ → (0 BernPoly (𝑋 + 1)) = 1)
101	5, 100	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 BernPoly (𝑋 + 1)) = 1)
102	101	oveq1d 7432	. . . . . . . . 9 ⊢ (𝜑 → ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1)))
103	102	oveq2d 7433	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1))))
104	103	oveq1d 7432	. . . . . . 7 ⊢ (𝜑 → (((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
105	99, 104	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
106		oveq1 7424	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
107	106, 96	oveq12d 7435	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)))
108	93, 107	oveq12d 7435	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))))
109	76, 41, 108	fsum1p 15843	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
110		bpoly0 16142	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
111	3, 110	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 BernPoly 𝑋) = 1)
112	111	oveq1d 7432	. . . . . . . . 9 ⊢ (𝜑 → ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1)))
113	112	oveq2d 7433	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1))))
114	113	oveq1d 7432	. . . . . . 7 ⊢ (𝜑 → (((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
115	109, 114	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
116	105, 115	oveq12d 7435	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))))
117		0z 12630	. . . . . . . . 9 ⊢ 0 ∈ ℤ
118		bccl 14390	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 0 ∈ ℤ) → (𝑁C0) ∈ ℕ₀)
119	2, 117, 118	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (𝑁C0) ∈ ℕ₀)
120	119	nn0cnd 12595	. . . . . . 7 ⊢ (𝜑 → (𝑁C0) ∈ ℂ)
121	21	subid1d 11586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 0) = 𝑁)
122	121, 1	eqeltrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 0) ∈ ℕ)
123	122	peano2nnd 12278	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 0) + 1) ∈ ℕ)
124	123	nnrecred 12315	. . . . . . . 8 ⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈ ℝ)
125	124	recnd 11265	. . . . . . 7 ⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈ ℂ)
126	120, 125	mulcld 11257	. . . . . 6 ⊢ (𝜑 → ((𝑁C0) · (1 / ((𝑁 − 0) + 1))) ∈ ℂ)
127		fzfid 14041	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...(𝑁 − 1)) ∈ Fin)
128		fzp1ss 13634	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → ((0 + 1)...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
129	117, 128	ax-mp 5	. . . . . . . . 9 ⊢ ((0 + 1)...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
130	129	sseli 3930	. . . . . . . 8 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ (0...(𝑁 − 1)))
131	130, 35	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
132	127, 131	fsumcl 15823	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
133	130, 41	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
134	127, 133	fsumcl 15823	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
135	126, 132, 134	pnpcand 11634	. . . . 5 ⊢ (𝜑 → ((((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))) − (((𝑁C0) · (1 / ((𝑁 − 0) + 1))) + Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
136		1zzd 12653	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
137		0zd 12631	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
138	1	nnzd 12645	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
139		2z 12654	. . . . . . . . 9 ⊢ 2 ∈ ℤ
140		zsubcl 12664	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑁 − 2) ∈ ℤ)
141	138, 139, 140	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 2) ∈ ℤ)
142		fzssp1 13626	. . . . . . . . . . 11 ⊢ (0...(𝑁 − 2)) ⊆ (0...((𝑁 − 2) + 1))
143		2cnd 12347	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
144	21, 143, 81	subsubd 11625	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − (2 − 1)) = ((𝑁 − 2) + 1))
145		2m1e1 12393	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
146	145	oveq2i 7428	. . . . . . . . . . . . 13 ⊢ (𝑁 − (2 − 1)) = (𝑁 − 1)
147	144, 146	eqtr3di 2812	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 2) + 1) = (𝑁 − 1))
148	147	oveq2d 7433	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 2) + 1)) = (0...(𝑁 − 1)))
149	142, 148	sseqtrid 3976	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 2)) ⊆ (0...(𝑁 − 1)))
150	149	sselda 3934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → 𝑚 ∈ (0...(𝑁 − 1)))
151	150, 52	syldan 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
152		oveq2 7425	. . . . . . . . 9 ⊢ (𝑚 = (𝑘 − 1) → (𝑁C𝑚) = (𝑁C(𝑘 − 1)))
153		oveq2 7425	. . . . . . . . 9 ⊢ (𝑚 = (𝑘 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑘 − 1)))
154	152, 153	oveq12d 7435	. . . . . . . 8 ⊢ (𝑚 = (𝑘 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
155	136, 137, 141, 151, 154	fsumshft 15870	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...((𝑁 − 2) + 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
156	147	oveq2d 7433	. . . . . . . 8 ⊢ (𝜑 → ((0 + 1)...((𝑁 − 2) + 1)) = ((0 + 1)...(𝑁 − 1)))
157	156	sumeq1d 15791	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...((𝑁 − 2) + 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
158	155, 157	eqtrd 2797	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
159		0p1e1 12389	. . . . . . . . . 10 ⊢ (0 + 1) = 1
160	159	oveq1i 7427	. . . . . . . . 9 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
161	160	eleq2i 2854	. . . . . . . 8 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) ↔ 𝑘 ∈ (1...(𝑁 − 1)))
162		fzssp1 13626	. . . . . . . . . . . . . 14 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
163	24	oveq2d 7433	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
164	162, 163	sseqtrid 3976	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
165	164	sselda 3934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ (1...𝑁))
166		bcm1k 14383	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘)))
167	165, 166	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘)))
168	1	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
169	168	nncnd 12277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℂ)
170		elfznn 13612	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
171	170	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
172	171	nncnd 12277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
173		1cnd 11230	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 1 ∈ ℂ)
174	169, 172, 173	subsubd 11625	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 − 𝑘) + 1))
175	174	oveq1d 7432	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − (𝑘 − 1)) / 𝑘) = (((𝑁 − 𝑘) + 1) / 𝑘))
176	175	oveq2d 7433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘)) = ((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)))
177	167, 176	eqtrd 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) = ((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)))
178		bpolydiflem.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) = (𝑘 · (𝑋↑(𝑘 − 1))))
179	178	oveq1d 7432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) / ((𝑁 − 𝑘) + 1)) = ((𝑘 · (𝑋↑(𝑘 − 1))) / ((𝑁 − 𝑘) + 1)))
180	161, 130	sylbir 238	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ (0...(𝑁 − 1)))
181	180, 19	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
182	180, 39	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ)
183	180, 32	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ)
184	180, 33	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0)
185	181, 182, 183, 184	divsubdird 12058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) − (𝑘 BernPoly 𝑋)) / ((𝑁 − 𝑘) + 1)) = (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))
186	3	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ)
187		nnm1nn0 12573	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
188	171, 187	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈ ℕ₀)
189	186, 188	expcld 14214	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑋↑(𝑘 − 1)) ∈ ℂ)
190	172, 189, 183, 184	div23d 12056	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 · (𝑋↑(𝑘 − 1))) / ((𝑁 − 𝑘) + 1)) = ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))
191	179, 185, 190	3eqtr3d 2805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))
192	177, 191	oveq12d 7435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C𝑘) · (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))))
193	180, 16	sylan2 605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ)
194	181, 183, 184	divcld 12019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
195	182, 183, 184	divcld 12019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
196	193, 194, 195	subdid 11698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C𝑘) · (((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) − ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
197	168	nnnn0d 12593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
198	188	nn0zd 12644	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈ ℤ)
199		bccl 14390	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑘 − 1) ∈ ℤ) → (𝑁C(𝑘 − 1)) ∈ ℕ₀)
200	197, 198, 199	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈ ℕ₀)
201	200	nn0cnd 12595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ)
202	171	nnne0d 12314	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ≠ 0)
203	183, 172, 202	divcld 12019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁 − 𝑘) + 1) / 𝑘) ∈ ℂ)
204	172, 183, 184	divcld 12019	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
205	204, 189	mulcld 11257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))) ∈ ℂ)
206	201, 203, 205	mulassd 11260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = ((𝑁C(𝑘 − 1)) · ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))))
207	183, 172, 184, 202	divcan6d 12038	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) = 1)
208	207	oveq1d 7432	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) · (𝑋↑(𝑘 − 1))) = (1 · (𝑋↑(𝑘 − 1))))
209	203, 204, 189	mulassd 11260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) · (𝑋↑(𝑘 − 1))) = ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))))
210	189	mullidd 11255	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (1 · (𝑋↑(𝑘 − 1))) = (𝑋↑(𝑘 − 1)))
211	208, 209, 210	3eqtr3d 2805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = (𝑋↑(𝑘 − 1)))
212	211	oveq2d 7433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C(𝑘 − 1)) · ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
213	206, 212	eqtrd 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
214	192, 196, 213	3eqtr3d 2805	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
215	161, 214	sylan2b 606	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
216	215	sumeq2dv 15793	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))(((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
217	127, 131, 133	fsumsub 15878	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))(((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
218	158, 216, 217	3eqtr2rd 2804	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)))
219	116, 135, 218	3eqtrd 2801	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)))
220	92, 219	oveq12d 7435	. . 3 ⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = ((Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))) − Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))))
221		fzfid 14041	. . . . 5 ⊢ (𝜑 → (0...(𝑁 − 2)) ∈ Fin)
222	221, 151	fsumcl 15823	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
223	3, 75	expcld 14214	. . . . 5 ⊢ (𝜑 → (𝑋↑(𝑁 − 1)) ∈ ℂ)
224	21, 223	mulcld 11257	. . . 4 ⊢ (𝜑 → (𝑁 · (𝑋↑(𝑁 − 1))) ∈ ℂ)
225	222, 224	pncan2d 11599	. . 3 ⊢ (𝜑 → ((Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))) − Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚))) = (𝑁 · (𝑋↑(𝑁 − 1))))
226	220, 225	eqtrd 2797	. 2 ⊢ (𝜑 → ((((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) − (Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))))) = (𝑁 · (𝑋↑(𝑁 − 1))))
227	10, 43, 226	3eqtrd 2801	1 ⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ⊆ wss 3902 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 − cmin 11469 / cdiv 11899 ℕcn 12261 2c2 12323 ℕ₀cn0 12532 ℤcz 12619 ℤ_≥cuz 12891 ...cfz 13565 ↑cexp 14129 Ccbc 14370 Σcsu 15777 BernPoly cbp 16138

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-bpoly 16139

This theorem is used by: bpolydif 16147

W3C validator