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Theorem bpolylem 15992

Description: Lemma for bpolyval 15993. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypotheses**
Ref	Expression
bpoly.1	⊢ 𝐺 = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
bpoly.2	⊢ 𝐹 = wrecs( < , ℕ₀, 𝐺)

**Assertion**
Ref	Expression
bpolylem	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))

Distinct variable groups: 𝑔,𝑘,𝑛,𝐹 𝑔,𝑁,𝑘,𝑛 𝑔,𝑋,𝑘,𝑛 Allowed substitution hints: 𝐺(𝑔,𝑘,𝑛)

Proof of Theorem bpolylem Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥↑𝑛) = (𝑋↑𝑛))
2	1	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
3	2	csbeq2dv 3901	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
4	3	mpteq2dv 5251	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))) = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))
5		bpoly.1	. . . . . . . 8 ⊢ 𝐺 = (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))
6	4, 5	eqtr4di 2791	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))) = 𝐺)
7		wrecseq3 8305	. . . . . . 7 ⊢ ((𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))) = 𝐺 → wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))) = wrecs( < , ℕ₀, 𝐺))
8	6, 7	syl 17	. . . . . 6 ⊢ (𝑥 = 𝑋 → wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))) = wrecs( < , ℕ₀, 𝐺))
9		bpoly.2	. . . . . 6 ⊢ 𝐹 = wrecs( < , ℕ₀, 𝐺)
10	8, 9	eqtr4di 2791	. . . . 5 ⊢ (𝑥 = 𝑋 → wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))))) = 𝐹)
11	10	fveq1d 6894	. . . 4 ⊢ (𝑥 = 𝑋 → (wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚) = (𝐹‘𝑚))
12		fveq2 6892	. . . 4 ⊢ (𝑚 = 𝑁 → (𝐹‘𝑚) = (𝐹‘𝑁))
13	11, 12	sylan9eqr 2795	. . 3 ⊢ ((𝑚 = 𝑁 ∧ 𝑥 = 𝑋) → (wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚) = (𝐹‘𝑁))
14		df-bpoly 15991	. . 3 ⊢ BernPoly = (𝑚 ∈ ℕ₀, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚))
15		fvex 6905	. . 3 ⊢ (𝐹‘𝑁) ∈ V
16	13, 14, 15	ovmpoa 7563	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = (𝐹‘𝑁))
17		ltweuz 13926	. . . . 5 ⊢ < We (ℤ_≥‘0)
18		nn0uz 12864	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
19		weeq2 5666	. . . . . 6 ⊢ (ℕ₀ = (ℤ_≥‘0) → ( < We ℕ₀ ↔ < We (ℤ_≥‘0)))
20	18, 19	ax-mp 5	. . . . 5 ⊢ ( < We ℕ₀ ↔ < We (ℤ_≥‘0))
21	17, 20	mpbir 230	. . . 4 ⊢ < We ℕ₀
22		nn0ex 12478	. . . . 5 ⊢ ℕ₀ ∈ V
23		exse 5640	. . . . 5 ⊢ (ℕ₀ ∈ V → < Se ℕ₀)
24	22, 23	ax-mp 5	. . . 4 ⊢ < Se ℕ₀
25	9	wfr2 8336	. . . 4 ⊢ ((( < We ℕ₀ ∧ < Se ℕ₀) ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) = (𝐺‘(𝐹 ↾ Pred( < , ℕ₀, 𝑁))))
26	21, 24, 25	mpanl12 701	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = (𝐺‘(𝐹 ↾ Pred( < , ℕ₀, 𝑁))))
27	26	adantr 482	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝐹‘𝑁) = (𝐺‘(𝐹 ↾ Pred( < , ℕ₀, 𝑁))))
28		prednn0 13625	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → Pred( < , ℕ₀, 𝑁) = (0...(𝑁 − 1)))
29	28	adantr 482	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Pred( < , ℕ₀, 𝑁) = (0...(𝑁 − 1)))
30	29	reseq2d 5982	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝐹 ↾ Pred( < , ℕ₀, 𝑁)) = (𝐹 ↾ (0...(𝑁 − 1))))
31	30	fveq2d 6896	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝐺‘(𝐹 ↾ Pred( < , ℕ₀, 𝑁))) = (𝐺‘(𝐹 ↾ (0...(𝑁 − 1)))))
32	9	wfrfun 8332	. . . . . . 7 ⊢ (( < We ℕ₀ ∧ < Se ℕ₀) → Fun 𝐹)
33	21, 24, 32	mp2an 691	. . . . . 6 ⊢ Fun 𝐹
34		ovex 7442	. . . . . 6 ⊢ (0...(𝑁 − 1)) ∈ V
35		resfunexg 7217	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (0...(𝑁 − 1)) ∈ V) → (𝐹 ↾ (0...(𝑁 − 1))) ∈ V)
36	33, 34, 35	mp2an 691	. . . . 5 ⊢ (𝐹 ↾ (0...(𝑁 − 1))) ∈ V
37		dmeq 5904	. . . . . . . . 9 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → dom 𝑔 = dom (𝐹 ↾ (0...(𝑁 − 1))))
38	9	wfr1 8335	. . . . . . . . . . . 12 ⊢ (( < We ℕ₀ ∧ < Se ℕ₀) → 𝐹 Fn ℕ₀)
39	21, 24, 38	mp2an 691	. . . . . . . . . . 11 ⊢ 𝐹 Fn ℕ₀
40		fz0ssnn0 13596	. . . . . . . . . . 11 ⊢ (0...(𝑁 − 1)) ⊆ ℕ₀
41		fnssres 6674	. . . . . . . . . . 11 ⊢ ((𝐹 Fn ℕ₀ ∧ (0...(𝑁 − 1)) ⊆ ℕ₀) → (𝐹 ↾ (0...(𝑁 − 1))) Fn (0...(𝑁 − 1)))
42	39, 40, 41	mp2an 691	. . . . . . . . . 10 ⊢ (𝐹 ↾ (0...(𝑁 − 1))) Fn (0...(𝑁 − 1))
43	42	fndmi 6654	. . . . . . . . 9 ⊢ dom (𝐹 ↾ (0...(𝑁 − 1))) = (0...(𝑁 − 1))
44	37, 43	eqtrdi 2789	. . . . . . . 8 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → dom 𝑔 = (0...(𝑁 − 1)))
45	44	fveq2d 6896	. . . . . . 7 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → (♯‘dom 𝑔) = (♯‘(0...(𝑁 − 1))))
46		fveq1 6891	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → (𝑔‘𝑘) = ((𝐹 ↾ (0...(𝑁 − 1)))‘𝑘))
47		fvres 6911	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((𝐹 ↾ (0...(𝑁 − 1)))‘𝑘) = (𝐹‘𝑘))
48	46, 47	sylan9eq 2793	. . . . . . . . . . 11 ⊢ ((𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑔‘𝑘) = (𝐹‘𝑘))
49	48	oveq1d 7424	. . . . . . . . . 10 ⊢ ((𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)) = ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))
50	49	oveq2d 7425	. . . . . . . . 9 ⊢ ((𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))) = ((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1))))
51	44, 50	sumeq12rdv 15653	. . . . . . . 8 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1))))
52	51	oveq2d 7425	. . . . . . 7 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → ((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))))
53	45, 52	csbeq12dv 3903	. . . . . 6 ⊢ (𝑔 = (𝐹 ↾ (0...(𝑁 − 1))) → ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))))
54		ovex 7442	. . . . . . 7 ⊢ ((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) ∈ V
55	54	csbex 5312	. . . . . 6 ⊢ ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) ∈ V
56	53, 5, 55	fvmpt 6999	. . . . 5 ⊢ ((𝐹 ↾ (0...(𝑁 − 1))) ∈ V → (𝐺‘(𝐹 ↾ (0...(𝑁 − 1)))) = ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))))
57	36, 56	ax-mp 5	. . . 4 ⊢ (𝐺‘(𝐹 ↾ (0...(𝑁 − 1)))) = ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1))))
58		nfcvd 2905	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → Ⅎ𝑛((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))))
59		oveq2 7417	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑋↑𝑛) = (𝑋↑𝑁))
60		oveq1 7416	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
61		oveq1 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
62	61	oveq1d 7424	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → ((𝑛 − 𝑘) + 1) = ((𝑁 − 𝑘) + 1))
63	62	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)) = ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))
64	60, 63	oveq12d 7427	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1))) = ((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1))))
65	64	sumeq2sdv 15650	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1))))
66	59, 65	oveq12d 7427	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))))
67	58, 66	csbiegf 3928	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ⦋𝑁 / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))))
68	67	adantr 482	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ⦋𝑁 / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))))
69		nn0z 12583	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
70		fz01en 13529	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (0...(𝑁 − 1)) ≈ (1...𝑁))
72		fzfi 13937	. . . . . . . . . 10 ⊢ (0...(𝑁 − 1)) ∈ Fin
73		fzfi 13937	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ Fin
74		hashen 14307	. . . . . . . . . 10 ⊢ (((0...(𝑁 − 1)) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘(0...(𝑁 − 1))) = (♯‘(1...𝑁)) ↔ (0...(𝑁 − 1)) ≈ (1...𝑁)))
75	72, 73, 74	mp2an 691	. . . . . . . . 9 ⊢ ((♯‘(0...(𝑁 − 1))) = (♯‘(1...𝑁)) ↔ (0...(𝑁 − 1)) ≈ (1...𝑁))
76	71, 75	sylibr 233	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0...(𝑁 − 1))) = (♯‘(1...𝑁)))
77		hashfz1 14306	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
78	76, 77	eqtrd 2773	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0...(𝑁 − 1))) = 𝑁)
79	78	adantr 482	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (♯‘(0...(𝑁 − 1))) = 𝑁)
80	79	csbeq1d 3898	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ⦋𝑁 / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))))
81		elfznn0 13594	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
82		simpr 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → 𝑋 ∈ ℂ)
83		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
84	11, 83	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑘 ∧ 𝑥 = 𝑋) → (wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚) = (𝐹‘𝑘))
85		fvex 6905	. . . . . . . . . . 11 ⊢ (𝐹‘𝑘) ∈ V
86	84, 14, 85	ovmpoa 7563	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑘 BernPoly 𝑋) = (𝐹‘𝑘))
87	81, 82, 86	syl2anr 598	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑘 BernPoly 𝑋) = (𝐹‘𝑘))
88	87	oveq1d 7424	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)) = ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))
89	88	oveq2d 7425	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = ((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1))))
90	89	sumeq2dv 15649	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1))))
91	90	oveq2d 7425	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝐹‘𝑘) / ((𝑁 − 𝑘) + 1)))))
92	68, 80, 91	3eqtr4d 2783	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → ⦋(♯‘(0...(𝑁 − 1))) / 𝑛⦌((𝑋↑𝑛) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑛C𝑘) · ((𝐹‘𝑘) / ((𝑛 − 𝑘) + 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
93	57, 92	eqtrid 2785	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝐺‘(𝐹 ↾ (0...(𝑁 − 1)))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
94	31, 93	eqtrd 2773	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝐺‘(𝐹 ↾ Pred( < , ℕ₀, 𝑁))) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))
95	16, 27, 94	3eqtrd 2777	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (𝑁 BernPoly 𝑋) = ((𝑋↑𝑁) − Σ𝑘 ∈ (0...(𝑁 − 1))((𝑁C𝑘) · ((𝑘 BernPoly 𝑋) / ((𝑁 − 𝑘) + 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⦋csb 3894 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 Se wse 5630 We wwe 5631 dom cdm 5677 ↾ cres 5679 Predcpred 6300 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 wrecscwrecs 8296 ≈ cen 8936 Fincfn 8939 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 − cmin 11444 / cdiv 11871 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 ↑cexp 14027 Ccbc 14262 ♯chash 14290 Σcsu 15632 BernPoly cbp 15990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-seq 13967 df-hash 14291 df-sum 15633 df-bpoly 15991

This theorem is referenced by: bpolyval 15993

W3C validator