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Theorem bpolydiflem 16109

Description: Lemma for bpolydif 16110. (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 12566	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3		bpolydiflem.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ)
4		peano2cn 11383	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + 1) ∈ ℂ)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 + 1) ∈ ℂ)
6		bpolyval 16104	. . . 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 595	. . 3 ⊢ (𝜑 → (𝑁 BernPoly (𝑋 + 1)) = (((𝑋 + 1)↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)))))
8		bpolyval 16104	. . . 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 595	. . 3 ⊢ (𝜑 → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
10	7, 9	oveq12d 7430	. 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 14184	. . 3 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) ∈ ℂ)
12		fzfid 14011	. . . 4 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
13		elfzelz 13553	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ)
14		bccl 14360	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
15	2, 13, 14	syl2an 607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℕ₀)
16	15	nn0cnd 12568	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ)
17		elfznn0 13650	. . . . . . 7 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
18		bpolycl 16107	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑋 + 1) ∈ ℂ) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
19	17, 5, 18	syl2anr 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
20		fzssp1 13597	. . . . . . . . . . 11 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
21	1	nncnd 12250	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
22		ax-1cn 11159	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
23		npcan 11467	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
24	21, 22, 23	sylancl 597	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
25	24	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁))
26	20, 25	sseqtrid 3980	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
27	26	sselda 3938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ (0...𝑁))
28		fznn0sub 13586	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ ℕ₀)
29	27, 28	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 𝑘) ∈ ℕ₀)
30		nn0p1nn 12544	. . . . . . . 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 12250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ)
33	31	nnne0d 12287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0)
34	19, 32, 33	divcld 11992	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
35	16, 34	mulcld 11230	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
36	12, 35	fsumcl 15786	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
37	3, 2	expcld 14184	. . 3 ⊢ (𝜑 → (𝑋↑𝑁) ∈ ℂ)
38		bpolycl 16107	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑘 BernPoly 𝑋) ∈ ℂ)
39	17, 3, 38	syl2anr 608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ)
40	39, 32, 33	divcld 11992	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
41	16, 40	mulcld 11230	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
42	12, 41	fsumcl 15786	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
43	11, 36, 37, 42	sub4d 11619	. 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 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...𝑁))
45		bccl2 14361	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ)
46	45	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ)
47	46	nncnd 12250	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ)
48		elfznn0 13650	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ₀)
49		expcl 14117	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝑋↑𝑚) ∈ ℂ)
50	3, 48, 49	syl2an 607	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑋↑𝑚) ∈ ℂ)
51	47, 50	mulcld 11230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
52	44, 51	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
53	12, 52	fsumcl 15786	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
54		addcom 11397	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑋 + 1) = (1 + 𝑋))
55	3, 22, 54	sylancl 597	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 + 1) = (1 + 𝑋))
56	55	oveq1d 7427	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = ((1 + 𝑋)↑𝑁))
57		binom1p 15887	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
58	3, 2, 57	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝑋)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
59	56, 58	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)))
60		nn0uz 12901	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
61	2, 60	eleqtrdi 2873	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
62		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑁C𝑚) = (𝑁C𝑁))
63		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑋↑𝑚) = (𝑋↑𝑁))
64	62, 63	oveq12d 7430	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C𝑁) · (𝑋↑𝑁)))
65	61, 51, 64	fsumm1 15804	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (𝑋↑𝑚)) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + ((𝑁C𝑁) · (𝑋↑𝑁))))
66		bcnn 14350	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁C𝑁) = 1)
67	2, 66	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁C𝑁) = 1)
68	67	oveq1d 7427	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (1 · (𝑋↑𝑁)))
69	37	mullidd 11228	. . . . . . . . 9 ⊢ (𝜑 → (1 · (𝑋↑𝑁)) = (𝑋↑𝑁))
70	68, 69	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C𝑁) · (𝑋↑𝑁)) = (𝑋↑𝑁))
71	70	oveq2d 7428	. . . . . . 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 2802	. . . . . 6 ⊢ (𝜑 → ((𝑋 + 1)↑𝑁) = (Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑋↑𝑁)))
73	53, 37, 72	mvrraddd 11627	. . . . 5 ⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = Σ𝑚 ∈ (0...(𝑁 − 1))((𝑁C𝑚) · (𝑋↑𝑚)))
74		nnm1nn0 12546	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
75	1, 74	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
76	75, 60	eleqtrdi 2873	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘0))
77		oveq2 7420	. . . . . . 7 ⊢ (𝑚 = (𝑁 − 1) → (𝑁C𝑚) = (𝑁C(𝑁 − 1)))
78		oveq2 7420	. . . . . . 7 ⊢ (𝑚 = (𝑁 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑁 − 1)))
79	77, 78	oveq12d 7430	. . . . . 6 ⊢ (𝑚 = (𝑁 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))))
80	76, 52, 79	fsumm1 15804	. . . . 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 11203	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
82	21, 81, 81	subsub4d 11601	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
83		df-2 12304	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
84	83	oveq2i 7423	. . . . . . . . 9 ⊢ (𝑁 − 2) = (𝑁 − (1 + 1))
85	82, 84	eqtr4di 2816	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
86	85	oveq2d 7428	. . . . . . 7 ⊢ (𝜑 → (0...((𝑁 − 1) − 1)) = (0...(𝑁 − 2)))
87	86	sumeq1d 15753	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...((𝑁 − 1) − 1))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)))
88		bcnm1 14365	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁C(𝑁 − 1)) = 𝑁)
89	2, 88	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑁C(𝑁 − 1)) = 𝑁)
90	89	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((𝑁C(𝑁 − 1)) · (𝑋↑(𝑁 − 1))) = (𝑁 · (𝑋↑(𝑁 − 1))))
91	87, 90	oveq12d 7430	. . . . 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 2802	. . . 4 ⊢ (𝜑 → (((𝑋 + 1)↑𝑁) − (𝑋↑𝑁)) = (Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) + (𝑁 · (𝑋↑(𝑁 − 1)))))
93		oveq2 7420	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑁C𝑘) = (𝑁C0))
94		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑘 BernPoly (𝑋 + 1)) = (0 BernPoly (𝑋 + 1)))
95		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑁 − 𝑘) = (𝑁 − 0))
96	95	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((𝑁 − 𝑘) + 1) = ((𝑁 − 0) + 1))
97	94, 96	oveq12d 7430	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)))
98	93, 97	oveq12d 7430	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))))
99	76, 35, 98	fsum1p 15806	. . . . . . 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 16105	. . . . . . . . . . 11 ⊢ ((𝑋 + 1) ∈ ℂ → (0 BernPoly (𝑋 + 1)) = 1)
101	5, 100	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 BernPoly (𝑋 + 1)) = 1)
102	101	oveq1d 7427	. . . . . . . . 9 ⊢ (𝜑 → ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1)))
103	102	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly (𝑋 + 1)) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1))))
104	103	oveq1d 7427	. . . . . . 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 2798	. . . . . 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 7419	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
107	106, 96	oveq12d 7430	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)))
108	93, 107	oveq12d 7430	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))))
109	76, 41, 108	fsum1p 15806	. . . . . . 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 16105	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
111	3, 110	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 BernPoly 𝑋) = 1)
112	111	oveq1d 7427	. . . . . . . . 9 ⊢ (𝜑 → ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1)) = (1 / ((𝑁 − 0) + 1)))
113	112	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → ((𝑁C0) · ((0 BernPoly 𝑋) / ((𝑁 − 0) + 1))) = ((𝑁C0) · (1 / ((𝑁 − 0) + 1))))
114	113	oveq1d 7427	. . . . . . 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 2798	. . . . . 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 7430	. . . . 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 12603	. . . . . . . . 9 ⊢ 0 ∈ ℤ
118		bccl 14360	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 0 ∈ ℤ) → (𝑁C0) ∈ ℕ₀)
119	2, 117, 118	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → (𝑁C0) ∈ ℕ₀)
120	119	nn0cnd 12568	. . . . . . 7 ⊢ (𝜑 → (𝑁C0) ∈ ℂ)
121	21	subid1d 11559	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 0) = 𝑁)
122	121, 1	eqeltrd 2863	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 0) ∈ ℕ)
123	122	peano2nnd 12251	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 0) + 1) ∈ ℕ)
124	123	nnrecred 12288	. . . . . . . 8 ⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈ ℝ)
125	124	recnd 11238	. . . . . . 7 ⊢ (𝜑 → (1 / ((𝑁 − 0) + 1)) ∈ ℂ)
126	120, 125	mulcld 11230	. . . . . 6 ⊢ (𝜑 → ((𝑁C0) · (1 / ((𝑁 − 0) + 1))) ∈ ℂ)
127		fzfid 14011	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...(𝑁 − 1)) ∈ Fin)
128		fzp1ss 13605	. . . . . . . . . 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 3934	. . . . . . . 8 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ (0...(𝑁 − 1)))
131	130, 35	sylan2 604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
132	127, 131	fsumcl 15786	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
133	130, 41	sylan2 604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
134	127, 133	fsumcl 15786	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) ∈ ℂ)
135	126, 132, 134	pnpcand 11607	. . . . 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 12626	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
137		0zd 12604	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
138	1	nnzd 12618	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
139		2z 12627	. . . . . . . . 9 ⊢ 2 ∈ ℤ
140		zsubcl 12637	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑁 − 2) ∈ ℤ)
141	138, 139, 140	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 2) ∈ ℤ)
142		fzssp1 13597	. . . . . . . . . . 11 ⊢ (0...(𝑁 − 2)) ⊆ (0...((𝑁 − 2) + 1))
143		2cnd 12320	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℂ)
144	21, 143, 81	subsubd 11598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − (2 − 1)) = ((𝑁 − 2) + 1))
145		2m1e1 12366	. . . . . . . . . . . . . 14 ⊢ (2 − 1) = 1
146	145	oveq2i 7423	. . . . . . . . . . . . 13 ⊢ (𝑁 − (2 − 1)) = (𝑁 − 1)
147	144, 146	eqtr3di 2813	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 2) + 1) = (𝑁 − 1))
148	147	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑁 − 2) + 1)) = (0...(𝑁 − 1)))
149	142, 148	sseqtrid 3980	. . . . . . . . . 10 ⊢ (𝜑 → (0...(𝑁 − 2)) ⊆ (0...(𝑁 − 1)))
150	149	sselda 3938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → 𝑚 ∈ (0...(𝑁 − 1)))
151	150, 52	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑁 − 2))) → ((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
152		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = (𝑘 − 1) → (𝑁C𝑚) = (𝑁C(𝑘 − 1)))
153		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = (𝑘 − 1) → (𝑋↑𝑚) = (𝑋↑(𝑘 − 1)))
154	152, 153	oveq12d 7430	. . . . . . . 8 ⊢ (𝑚 = (𝑘 − 1) → ((𝑁C𝑚) · (𝑋↑𝑚)) = ((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
155	136, 137, 141, 151, 154	fsumshft 15833	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...((𝑁 − 2) + 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
156	147	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → ((0 + 1)...((𝑁 − 2) + 1)) = ((0 + 1)...(𝑁 − 1)))
157	156	sumeq1d 15753	. . . . . . 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 2798	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) = Σ𝑘 ∈ ((0 + 1)...(𝑁 − 1))((𝑁C(𝑘 − 1)) · (𝑋↑(𝑘 − 1))))
159		0p1e1 12362	. . . . . . . . . 10 ⊢ (0 + 1) = 1
160	159	oveq1i 7422	. . . . . . . . 9 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
161	160	eleq2i 2855	. . . . . . . 8 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) ↔ 𝑘 ∈ (1...(𝑁 − 1)))
162		fzssp1 13597	. . . . . . . . . . . . . 14 ⊢ (1...(𝑁 − 1)) ⊆ (1...((𝑁 − 1) + 1))
163	24	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
164	162, 163	sseqtrid 3980	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
165	164	sselda 3938	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ (1...𝑁))
166		bcm1k 14353	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
169	168	nncnd 12250	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℂ)
170		elfznn 13583	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(𝑁 − 1)) → 𝑘 ∈ ℕ)
171	170	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℕ)
172	171	nncnd 12250	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ∈ ℂ)
173		1cnd 11203	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 1 ∈ ℂ)
174	169, 172, 173	subsubd 11598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 − 𝑘) + 1))
175	174	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − (𝑘 − 1)) / 𝑘) = (((𝑁 − 𝑘) + 1) / 𝑘))
176	175	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁C(𝑘 − 1)) · ((𝑁 − (𝑘 − 1)) / 𝑘)) = ((𝑁C(𝑘 − 1)) · (((𝑁 − 𝑘) + 1) / 𝑘)))
177	167, 176	eqtrd 2798	. . . . . . . . . 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 7427	. . . . . . . . . . 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 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly (𝑋 + 1)) ∈ ℂ)
182	180, 39	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) ∈ ℂ)
183	180, 32	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ∈ ℂ)
184	180, 33	sylan2 604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑁 − 𝑘) + 1) ≠ 0)
185	181, 182, 183, 184	divsubdird 12031	. . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑋 ∈ ℂ)
187		nnm1nn0 12546	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
188	171, 187	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈ ℕ₀)
189	186, 188	expcld 14184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑋↑(𝑘 − 1)) ∈ ℂ)
190	172, 189, 183, 184	div23d 12029	. . . . . . . . . . 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 2806	. . . . . . . . . 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 7430	. . . . . . . . 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 604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C𝑘) ∈ ℂ)
194	181, 183, 184	divcld 11992	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly (𝑋 + 1)) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
195	182, 183, 184	divcld 11992	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
196	193, 194, 195	subdid 11671	. . . . . . . . 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 12566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
198	188	nn0zd 12617	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 − 1) ∈ ℤ)
199		bccl 14360	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑘 − 1) ∈ ℤ) → (𝑁C(𝑘 − 1)) ∈ ℕ₀)
200	197, 198, 199	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈ ℕ₀)
201	200	nn0cnd 12568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ)
202	171	nnne0d 12287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → 𝑘 ≠ 0)
203	183, 172, 202	divcld 11992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (((𝑁 − 𝑘) + 1) / 𝑘) ∈ ℂ)
204	172, 183, 184	divcld 11992	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (𝑘 / ((𝑁 − 𝑘) + 1)) ∈ ℂ)
205	204, 189	mulcld 11230	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1))) ∈ ℂ)
206	201, 203, 205	mulassd 11233	. . . . . . . . . 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 12011	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · (𝑘 / ((𝑁 − 𝑘) + 1))) = 1)
208	207	oveq1d 7427	. . . . . . . . . . . 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 11233	. . . . . . . . . . . 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 11228	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → (1 · (𝑋↑(𝑘 − 1))) = (𝑋↑(𝑘 − 1)))
211	208, 209, 210	3eqtr3d 2806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 − 1))) → ((((𝑁 − 𝑘) + 1) / 𝑘) · ((𝑘 / ((𝑁 − 𝑘) + 1)) · (𝑋↑(𝑘 − 1)))) = (𝑋↑(𝑘 − 1)))
212	211	oveq2d 7428	. . . . . . . . . 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 2798	. . . . . . . . 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 2806	. . . . . . . 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 605	. . . . . . 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 15755	. . . . . 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 15841	. . . . . 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 2805	. . . . 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 2802	. . . 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 7430	. . 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 14011	. . . . 5 ⊢ (𝜑 → (0...(𝑁 − 2)) ∈ Fin)
222	221, 151	fsumcl 15786	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑁 − 2))((𝑁C𝑚) · (𝑋↑𝑚)) ∈ ℂ)
223	3, 75	expcld 14184	. . . . 5 ⊢ (𝜑 → (𝑋↑(𝑁 − 1)) ∈ ℂ)
224	21, 223	mulcld 11230	. . . 4 ⊢ (𝜑 → (𝑁 · (𝑋↑(𝑁 − 1))) ∈ ℂ)
225	222, 224	pncan2d 11572	. . 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 2798	. 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 2802	1 ⊢ (𝜑 → ((𝑁 BernPoly (𝑋 + 1)) − (𝑁 BernPoly 𝑋)) = (𝑁 · (𝑋↑(𝑁 − 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ⊆ wss 3906 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 / cdiv 11872 ℕcn 12234 2c2 12296 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536 ↑cexp 14099 Ccbc 14340 Σcsu 15739 BernPoly cbp 16101

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-bpoly 16102

This theorem is referenced by: bpolydif 16110

W3C validator