Theorem dvnmul 46922

Description: Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
dvnmul.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnmul.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
dvnmul.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvnmul.cc	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
dvnmul.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
dvnmulf	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴)
dvnmul.f	⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)
dvnmul.dvnf	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
dvnmul.dvng	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
dvnmul.c	⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
dvnmul.d	⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))

Assertion

Ref	Expression
dvnmul	⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑥,𝐶   𝑥,𝐷   𝑘,𝐹   𝑘,𝐺   𝑘,𝑁,𝑥   𝑆,𝑘,𝑥   𝑘,𝑋,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑘)   𝐷(𝑘)   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem dvnmul

Dummy variables 𝑖 𝑚 𝑛 ℎ 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝜑 → 𝜑)
2		dvnmul.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
3		nn0uz 12996	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
4	2, 3	eleqtrdi 2871	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
5		eluzfz2 13658	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → 𝑁 ∈ (0...𝑁))
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
7		eleq1 2849	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8		fveq2 6883	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9		oveq2 7426	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
10	9	sumeq1d 15860	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
11		oveq1 7425	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12		fvoveq1 7441	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝐷‘(𝑛 − 𝑘)) = (𝐷‘(𝑁 − 𝑘)))
13	12	fveq1d 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → ((𝐷‘(𝑛 − 𝑘))‘𝑥) = ((𝐷‘(𝑁 − 𝑘))‘𝑥))
14	13	oveq2d 7434	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))
15	11, 14	oveq12d 7436	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
16	15	sumeq2sdv 15863	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
17	10, 16	eqtrd 2796	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
18	17	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))
19	8, 18	eqeq12d 2777	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))
20	19	imbi2d 343	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))
21	7, 20	imbi12d 347	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) ↔ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))))
22		fveq2 6883	. . . . . . 7 ⊢ (𝑚 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0))
23		simpl 488	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 0)
24	23	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...0))
25		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
26	25	oveq1d 7433	. . . . . . . . . 10 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
27	25	fvoveq1d 7440	. . . . . . . . . . . 12 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(0 − 𝑘)))
28	27	fveq1d 6885	. . . . . . . . . . 11 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
29	28	oveq2d 7434	. . . . . . . . . 10 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
30	26, 29	oveq12d 7436	. . . . . . . . 9 ⊢ (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
31	24, 30	sumeq12rdv 15866	. . . . . . . 8 ⊢ ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
32	31	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
33	22, 32	eqeq12d 2777	. . . . . 6 ⊢ (𝑚 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
34	33	imbi2d 343	. . . . 5 ⊢ (𝑚 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))))
35		fveq2 6883	. . . . . . 7 ⊢ (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
36		simpl 488	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑖)
37	36	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑖))
38		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
39	38	oveq1d 7433	. . . . . . . . . 10 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
40	38	fvoveq1d 7440	. . . . . . . . . . . 12 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑖 − 𝑘)))
41	40	fveq1d 6885	. . . . . . . . . . 11 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − 𝑘))‘𝑥))
42	41	oveq2d 7434	. . . . . . . . . 10 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
43	39, 42	oveq12d 7436	. . . . . . . . 9 ⊢ (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))
44	37, 43	sumeq12rdv 15866	. . . . . . . 8 ⊢ ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))
45	44	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑚 = 𝑖 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
46	35, 45	eqeq12d 2777	. . . . . 6 ⊢ (𝑚 = 𝑖 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
47	46	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑖 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))))
48		fveq2 6883	. . . . . . 7 ⊢ (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
49		simpl 488	. . . . . . . . . 10 ⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → 𝑚 = (𝑖 + 1))
50	49	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
51		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
52	51	oveq1d 7433	. . . . . . . . . 10 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
53	51	fvoveq1d 7440	. . . . . . . . . . . 12 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
54	53	fveq1d 6885	. . . . . . . . . . 11 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
55	54	oveq2d 7434	. . . . . . . . . 10 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
56	52, 55	oveq12d 7436	. . . . . . . . 9 ⊢ (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
57	50, 56	sumeq12rdv 15866	. . . . . . . 8 ⊢ ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
58	57	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑚 = (𝑖 + 1) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
59	48, 58	eqeq12d 2777	. . . . . 6 ⊢ (𝑚 = (𝑖 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
60	59	imbi2d 343	. . . . 5 ⊢ (𝑚 = (𝑖 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
61		fveq2 6883	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
62		simpl 488	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑛)
63	62	oveq2d 7434	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑛))
64		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
65	64	oveq1d 7433	. . . . . . . . . 10 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
66	64	fvoveq1d 7440	. . . . . . . . . . . 12 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑛 − 𝑘)))
67	66	fveq1d 6885	. . . . . . . . . . 11 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑛 − 𝑘))‘𝑥))
68	67	oveq2d 7434	. . . . . . . . . 10 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))
69	65, 68	oveq12d 7436	. . . . . . . . 9 ⊢ (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
70	63, 69	sumeq12rdv 15866	. . . . . . . 8 ⊢ ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
71	70	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))
72	61, 71	eqeq12d 2777	. . . . . 6 ⊢ (𝑚 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))
73	72	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))))
74		dvnmul.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
75		recnprss 26217	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
76	74, 75	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
77		dvnmul.a	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
78		dvnmul.cc	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
79	77, 78	mulcld 11322	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) ∈ ℂ)
80		restsspw 17595	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
81		dvnmul.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
82	80, 81	sselid 3929	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
83		elpwi 4564	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
84	82, 83	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
85		cnex 11274	. . . . . . . . . 10 ⊢ ℂ ∈ V
86	85	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ V)
87	79, 84, 86, 74	mptelpm 46160	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
88		dvn0 26237	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
89	76, 87, 88	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
90		0z 12697	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
91		fzsn 13693	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → (0...0) = {0})
92	90, 91	ax-mp 5	. . . . . . . . . . 11 ⊢ (0...0) = {0}
93	92	sumeq1i 15857	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
94	93	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
95		nfcvd 2924	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Ⅎ𝑘(𝐴 · 𝐵))
96		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
97		oveq2 7426	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
98		0nn0 12614	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
99		bcn0 14447	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
100	98, 99	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0C0) = 1
101	100	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (0C0) = 1)
102	97, 101	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
103	102	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
104		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
105	104	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = (𝐶‘0))
106		dvnmul.c	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
107		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
108	107	cbvmptv 5209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
109	106, 108	eqtri 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
110		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
111		eluzfz1 13657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑁))
112	4, 111	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ∈ (0...𝑁))
113		fvexd 6898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
114	109, 110, 112, 113	fvmptd3 7015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
115	114	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
116	105, 115	eqtrd 2796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘0))
117		dvnmulf	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴)
118	77, 84, 86, 74	mptelpm 46160	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
119	117, 118	eqeltrid 2865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆))
120		dvn0 26237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
121	76, 119, 120	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
122	121	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
123	116, 122	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = 𝐹)
124	123	fveq1d 6885	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥))
125	124	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥))
126		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
127	117	fvmpt2 7003	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝐹‘𝑥) = 𝐴)
128	126, 77, 127	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = 𝐴)
129	128	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐹‘𝑥) = 𝐴)
130	125, 129	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = 𝐴)
131		oveq2 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
132		0m0e0 12454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 − 0) = 0
133	132	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (0 − 0) = 0)
134	131, 133	eqtrd 2796	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
135	134	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
136	135	fveq1d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
137	136	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
138	137	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
139		dvnmul.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
140		fveq2 6883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
141	140	cbvmptv 5209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
142	139, 141	eqtri 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
143	142	fveq1i 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)
144	143	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0))
145		eqidd 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
146		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
147	146	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
148		dvnmul.f	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)
149	78, 84, 86, 74	mptelpm 46160	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (ℂ ↑pm 𝑆))
150	148, 149	eqeltrid 2865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑pm 𝑆))
151		dvn0 26237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
152	76, 150, 151	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
153	152	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
154	147, 153	eqtrd 2796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
155	148	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵))
156		mptexg 7225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 ∈ 𝒫 𝑆 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V)
157	82, 156	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V)
158	155, 157	eqeltrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 ∈ V)
159	145, 154, 112, 158	fvmptd 6999	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
160	144, 159	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐷‘0) = 𝐺)
161	160	fveq1d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥))
162	161	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥))
163	155, 78	fvmpt2d 7005	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = 𝐵)
164	163	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐺‘𝑥) = 𝐵)
165	138, 162, 164	3eqtrd 2800	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
166	130, 165	oveq12d 7436	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
167	103, 166	oveq12d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
168	79	mullidd 11320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
169	168	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
170	167, 169	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
171		0re 11303	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
172	171	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
173	95, 96, 170, 172, 79	sumsnd 46012	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
174	94, 173	eqtr2d 2797	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
175	174	mpteq2dva 5198	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
176	89, 175	eqtrd 2796	. . . . . 6 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
177	176	a1i 11	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
178		simp3 1156	. . . . . . 7 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑)
179		simp1 1154	. . . . . . 7 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝑖 ∈ (0..^𝑁))
180		simp2 1155	. . . . . . . 8 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
181		pm3.35 815	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
182	178, 180, 181	syl2anc 596	. . . . . . 7 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
183	76	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
184	87	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
185		elfzonn0 13835	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
186	185	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
187		dvnp1 26238	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑖 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
188	183, 184, 186, 187	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
189	188	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
190		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
191	190	oveq2d 7434	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
192		eqid 2761	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
193		eqid 2761	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
194	74	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
195	81	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
196		fzfid 14109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
197	185	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
198		elfzelz 13649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
199	198	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
200	197, 199	bccld 46299	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
201	200	nn0cnd 12662	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
202	201	adantll 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
203	202	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ)
204		simpll 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
205		0zd 12698	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
206		elfzoel2 13785	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
207	206	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
208		elfzle1 13653	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
209	208	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
210	199	zred 12796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
211	206	zred 12796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
212	211	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
213	185	nn0red 12661	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
214	213	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
215		elfzle2 13654	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ 𝑖)
216	215	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑖)
217		elfzolt2 13796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
218	217	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
219	210, 214, 212, 216, 218	lelttrd 11461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
220	210, 212, 219	ltled 11451	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑁)
221	205, 207, 199, 209, 220	elfzd 13640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
222	221	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
223		dvnmul.dvnf	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
224	106	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
225		fvexd 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
226	224, 225	fvmpt2d 7005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
227	226	feq1d 6689	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
228	223, 227	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ)
229	204, 222, 228	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘):𝑋⟶ℂ)
230	229	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐶‘𝑘):𝑋⟶ℂ)
231		simp3 1156	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
232	230, 231	ffvelcdmd 7083	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
233	185	nn0zd 12711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
234	233	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
235	234, 199	zsubcld 12801	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℤ)
236		elfzel2 13647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
237	236	zred 12796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
238	198	zred 12796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
239	237, 238	subge0d 11899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖 − 𝑘) ↔ 𝑘 ≤ 𝑖))
240	215, 239	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖 − 𝑘))
241	240	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖 − 𝑘))
242	214, 210	resubcld 11737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℝ)
243	212, 210	resubcld 11737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ∈ ℝ)
244	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
245	212, 244	jca 521	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
246		resubcl 11615	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
247	245, 246	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
248	214, 212, 210, 218	ltsub1dd 11921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 𝑘))
249	244, 210, 212, 209	lesub2dd 11926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ (𝑁 − 0))
250	242, 243, 247, 248, 249	ltletrd 11463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 0))
251	211	recnd 11330	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
252	251	subid1d 11651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
253	252	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
254	250, 253	breqtrd 5131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < 𝑁)
255	242, 212, 254	ltled 11451	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ≤ 𝑁)
256	205, 207, 235, 241, 255	elfzd 13640	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁))
257	256	adantll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁))
258		ovex 7451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 − 𝑘) ∈ V
259		eleq1 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑖 − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖 − 𝑘) ∈ (0...𝑁)))
260	259	anbi2d 642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑖 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁))))
261		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
262	261	feq1d 6689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑖 − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ))
263	260, 262	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑖 − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)))
264		nfv 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
265		eleq1 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
266	265	anbi2d 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑁))))
267		fveq2 6883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
268	267	feq1d 6689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
269	266, 268	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
270		dvnmul.dvng	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
271	264, 269, 270	chvarfv 2277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
272	258, 263, 271	vtocl 3521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)
273	204, 257, 272	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)
274		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
275		fvexd 6898	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V)
276	142, 274, 256, 275	fvmptd3 7015	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
277	276	adantll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
278	277	feq1d 6689	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ))
279	273, 278	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ)
280	279	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ)
281	280, 231	ffvelcdmd 7083	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ)
282	232, 281	mulcld 11322	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
283	203, 282	mulcld 11322	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ)
284	203	3expa 1136	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ)
285	234	peano2zd 12799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
286	285, 199	zsubcld 12801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
287		peano2re 11476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
288	237, 287	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
289		peano2re 11476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
290	238, 289	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
291	238	ltp1d 12240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
292		1red 11302	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
293	238, 237, 292, 215	leadd1dd 11923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
294	238, 290, 288, 291, 293	ltletrd 11463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
295	238, 288, 294	ltled 11451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
296	295	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
297	214, 287	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
298	297, 210	subge0d 11899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
299	296, 298	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
300	297, 210	resubcld 11737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
301		elfzop1le2 13800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
302	301	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
303	297, 212, 210, 302	lesub1dd 11925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘))
304	249, 253	breqtrd 5131	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ 𝑁)
305	300, 243, 212, 303, 304	letrd 11460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
306	205, 207, 286, 299, 305	elfzd 13640	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
307	306	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
308		ovex 7451	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 + 1) − 𝑘) ∈ V
309		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
310	309	anbi2d 642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
311		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
312	311	feq1d 6689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
313	310, 312	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
314	308, 313, 271	vtocl 3521	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
315	204, 307, 314	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
316	142	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
317		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
318	317	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
319		fvexd 6898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
320	316, 318, 307, 319	fvmptd 6999	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
321	320	feq1d 6689	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
322	315, 321	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
323	322	ffvelcdmda 7082	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
324	232	3expa 1136	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
325	323, 324	mulcomd 11323	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
326	325	oveq2d 7434	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) = ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
327	199	peano2zd 12799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
328	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
329	328, 238, 290, 208, 291	lelttrd 11461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
330	328, 290, 329	ltled 11451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
331	330	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
332	210, 289	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
333	293	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
334	332, 297, 212, 333, 302	letrd 11460	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
335	205, 207, 327, 331, 334	elfzd 13640	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
336	335	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
337		ovex 7451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 + 1) ∈ V
338		eleq1 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
339	338	anbi2d 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
340		fveq2 6883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝑘 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑘 + 1)))
341	340	feq1d 6689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑘 + 1) → ((𝐶‘𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
342	339, 341	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
343		nfv 1947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (0...𝑁))
344		nfmpt1 5204	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
345	106, 344	nfcxfr 2921	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘𝐶
346		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘𝑗
347	345, 346	nffv 6893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘(𝐶‘𝑗)
348		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘𝑋
349		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘ℂ
350	347, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝐶‘𝑗):𝑋⟶ℂ
351	343, 350	nfim 1929	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)
352		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
353	352	feq1d 6689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑗 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘𝑗):𝑋⟶ℂ))
354	266, 353	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)))
355	351, 354, 228	chvarfv 2277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)
356	337, 342, 355	vtocl 3521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
357	204, 336, 356	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
358	357	ffvelcdmda 7082	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
359	281	3expa 1136	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ)
360	358, 359	mulcld 11322	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
361	323, 324	mulcld 11322	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) ∈ ℂ)
362	360, 361	addcld 11321	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) ∈ ℂ)
363	326, 362	eqeltrrd 2862	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
364	284, 363	mulcld 11322	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
365	364	3impa 1127	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
366	204, 74	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
367	171	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
368	204, 81	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
369	366, 368, 202	dvmptconst 46894	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑖C𝑘))) = (𝑥 ∈ 𝑋 ↦ 0))
370	282	3expa 1136	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
371	204, 222, 226	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
372	371	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶‘𝑘))
373	229	feqmptd 6951	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)))
374	372, 373	eqtr2d 2797	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
375	374	oveq2d 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
376	366, 75	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
377	204, 119	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
378		elfznn0 13747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
379	378	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
380		dvnp1 26238	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
381	376, 377, 379, 380	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
382	381	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
383		fveq2 6883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
384		fvexd 6898	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
385	109, 383, 336, 384	fvmptd3 7015	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
386	385	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
387	357	feqmptd 6951	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
388	386, 387	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
389	375, 382, 388	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
390	277	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − 𝑘)))
391	279	feqmptd 6951	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
392	390, 391	eqtr2d 2797	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
393	392	oveq2d 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
394	204, 150	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
395		fznn0sub 13683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝑖) → (𝑖 − 𝑘) ∈ ℕ0)
396	395	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℕ0)
397		dvnp1 26238	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖 − 𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
398	376, 394, 396, 397	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
399	398	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)))
400	214	recnd 11330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
401		1cnd 11295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
402	210	recnd 11330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
403	400, 401, 402	addsubd 11683	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖 − 𝑘) + 1))
404	403	eqcomd 2767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 − 𝑘) + 1) = ((𝑖 + 1) − 𝑘))
405	404	fveq2d 6887	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
406	405	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
407	320	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
408	322	feqmptd 6951	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
409	406, 407, 408	3eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
410	393, 399, 409	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
411	366, 324, 358, 389, 359, 323, 410	dvmptmul 26274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)))))
412	366, 284, 367, 369, 370, 362, 411	dvmptmul 26274	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)))))
413	370	mul02d 11501	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = 0)
414	326	oveq1d 7433	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)))
415	363, 284	mulcomd 11323	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
416	414, 415	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
417	413, 416	oveq12d 7436	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
418	364	addlidd 11504	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
419	417, 418	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
420	419	mpteq2dva 5198	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
421	412, 420	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
422	192, 193, 194, 195, 196, 283, 365, 421	dvmptfsum 26288	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
423	202	adantlr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
424	360	an32s 665	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
425		anass 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)))
426		ancom 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))
427	426	anbi2i 635	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))))
428		anass 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))))
429	428	bicomi 227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
430	427, 429	bitri 278	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
431	425, 430	bitri 278	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
432	431	imbi1i 352	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ))
433	324, 432	mpbi 233	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
434	431	imbi1i 352	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ))
435	323, 434	mpbi 233	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
436	433, 435	mulcld 11322	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
437	423, 424, 436	adddid 11326	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
438	437	sumeq2dv 15862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
439	196	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...𝑖) ∈ Fin)
440	423, 424	mulcld 11322	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ)
441	423, 436	mulcld 11322	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
442	439, 440, 441	fsumadd 15899	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
443		oveq2 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → (𝑖C𝑘) = (𝑖Cℎ))
444		fvoveq1 7441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝐶‘(𝑘 + 1)) = (𝐶‘(ℎ + 1)))
445	444	fveq1d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘(ℎ + 1))‘𝑥))
446		oveq2 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = ℎ → (𝑖 − 𝑘) = (𝑖 − ℎ))
447	446	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = ℎ → (𝐷‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − ℎ)))
448	447	fveq1d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → ((𝐷‘(𝑖 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − ℎ))‘𝑥))
449	445, 448	oveq12d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
450	443, 449	oveq12d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = ℎ → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))))
451		nfcv 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎℎ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
452		nfcv 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝑖Cℎ)
453		nfcv 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 ·
454		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘(ℎ + 1)
455	345, 454	nffv 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝐶‘(ℎ + 1))
456		nfcv 2923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘𝑥
457	455, 456	nffv 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝐶‘(ℎ + 1))‘𝑥)
458		nfmpt1 5204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
459	139, 458	nfcxfr 2921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘𝐷
460		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘(𝑖 − ℎ)
461	459, 460	nffv 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(𝐷‘(𝑖 − ℎ))
462	461, 456	nffv 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝐷‘(𝑖 − ℎ))‘𝑥)
463	457, 453, 462	nfov 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))
464	452, 453, 463	nfov 7448	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
465	450, 451, 464	cbvsum 15855	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
466	465	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))))
467		1zzd 12720	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℤ)
468	90	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℤ)
469	233	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ ℤ)
470		nfv 1947	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)
471		nfcv 2923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘ℎ
472		nfcv 2923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(0...𝑖)
473	471, 472	nfel 2937	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘 ℎ ∈ (0...𝑖)
474	470, 473	nfan 1932	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖))
475	464, 349	nfel 2937	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ
476	474, 475	nfim 1929	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)
477		eleq1 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ℎ → (𝑘 ∈ (0...𝑖) ↔ ℎ ∈ (0...𝑖)))
478	477	anbi2d 642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖))))
479	450	eleq1d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ℎ → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ))
480	478, 479	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = ℎ → (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)))
481	476, 480, 440	chvarfv 2277	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)
482		oveq2 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑗 − 1) → (𝑖Cℎ) = (𝑖C(𝑗 − 1)))
483		fvoveq1 7441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑗 − 1) → (𝐶‘(ℎ + 1)) = (𝐶‘((𝑗 − 1) + 1)))
484	483	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑗 − 1) → ((𝐶‘(ℎ + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
485		oveq2 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑗 − 1) → (𝑖 − ℎ) = (𝑖 − (𝑗 − 1)))
486	485	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑗 − 1) → (𝐷‘(𝑖 − ℎ)) = (𝐷‘(𝑖 − (𝑗 − 1))))
487	486	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑗 − 1) → ((𝐷‘(𝑖 − ℎ))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
488	484, 487	oveq12d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ = (𝑗 − 1) → (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
489	482, 488	oveq12d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑗 − 1) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
490	467, 468, 469, 481, 489	fsumshft 15939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
491	466, 490	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
492		0p1e1 12456	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
493	492	oveq1i 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
494	493	sumeq1i 15857	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
495	494	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
496		elfzelz 13649	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
497	496	zcnd 12797	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
498		1cnd 11295	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
499	497, 498	npcand 11666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
500	499	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶‘𝑗))
501	500	fveq1d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥))
502	501	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥))
503	213	recnd 11330	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
504	503	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
505	497	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
506	498	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
507	504, 505, 506	subsub3d 11692	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
508	507	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
509	508	fveq1d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
510	502, 509	oveq12d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) = (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
511	510	oveq2d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
512	511	sumeq2dv 15862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
513	512	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
514		nfv 1947	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)
515		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
516		fzfid 14109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...(𝑖 + 1)) ∈ Fin)
517	185	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
518	496	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
519		1zzd 12720	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
520	518, 519	zsubcld 12801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
521	517, 520	bccld 46299	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
522	521	nn0cnd 12662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
523	522	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
524	523	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
525	1	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
526		0zd 12698	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
527	206	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
528	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
529	496	zred 12796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
530		1red 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
531		0lt1 11831	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 < 1
532	531	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
533		elfzle1 13653	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
534	528, 530, 529, 532, 533	ltletrd 11463	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
535	528, 529, 534	ltled 11451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
536	535	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
537	529	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
538	213	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
539		1red 11302	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
540	538, 539	readdcld 11331	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
541	211	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
542		elfzle2 13654	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
543	542	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
544	301	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
545	537, 540, 541, 543, 544	letrd 11460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ 𝑁)
546	526, 527, 518, 536, 545	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
547	546	adantll 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
548	525, 547, 355	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ)
549	548	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ)
550		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥 ∈ 𝑋)
551	549, 550	ffvelcdmd 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ)
552	233	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
553	552	peano2zd 12799	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
554	553, 518	zsubcld 12801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
555	540, 537	subge0d 11899	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
556	543, 555	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
557	540, 537	resubcld 11737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
558	557	leidd 11875	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
559	529, 534	elrpd 13154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
560	559	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
561	540, 560	ltsubrpd 13189	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
562	557, 540, 541, 561, 544	ltletrd 11463	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
563	557, 557, 541, 558, 562	lelttrd 11461	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
564	557, 541, 563	ltled 11451	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
565	526, 527, 554, 556, 564	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
566	565	adantll 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
567		nfv 1947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
568		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑘((𝑖 + 1) − 𝑗)
569	459, 568	nffv 6893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗))
570	569, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
571	567, 570	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
572		ovex 7451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 + 1) − 𝑗) ∈ V
573		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
574	573	anbi2d 642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
575		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷‘𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
576	575	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
577	574, 576	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
578	139	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
579		fvexd 6898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
580	578, 579	fvmpt2d 7005	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
581	580	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
582	270, 581	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ)
583	571, 572, 577, 582	vtoclf 3526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
584	525, 566, 583	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
585	584	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
586	585, 550	ffvelcdmd 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
587	551, 586	mulcld 11322	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
588	524, 587	mulcld 11322	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
589		1zzd 12720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
590	233	peano2zd 12799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
591	492	eqcomi 2770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 = (0 + 1)
592	591	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
593	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
594		1red 11302	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
595	185	nn0ge0d 12663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
596	593, 213, 594, 595	leadd1dd 11923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
597	592, 596	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
598	589, 590, 597	3jca 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
599		eluz2 12964	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 + 1) ∈ (ℤ≥‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
600	598, 599	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ≥‘1))
601		eluzfz2 13658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 + 1) ∈ (ℤ≥‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
602	600, 601	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
603	602	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
604		oveq1 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
605	604	oveq2d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
606		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑖 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑖 + 1)))
607	606	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑖 + 1) → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
608		oveq2 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
609	608	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
610	609	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
611	607, 610	oveq12d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑖 + 1) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
612	605, 611	oveq12d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑖 + 1) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
613	514, 515, 516, 588, 603, 612	fsumsplit1 15904	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
614		1cnd 11295	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
615	503, 614	pncand 11663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
616	615	oveq2d 7434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
617		bcnn 14449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
618	185, 617	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
619	616, 618	eqtrd 2796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
620	503, 614	addcld 11321	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
621	620	subidd 11650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
622	621	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
623	622	fveq1d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
624	623	oveq2d 7434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
625	619, 624	oveq12d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))))
626	625	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))))
627		simpl 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝜑)
628		fzofzp1 13892	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
629	628	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
630		nfv 1947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
631		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑘(𝑖 + 1)
632	345, 631	nffv 6893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝐶‘(𝑖 + 1))
633	632, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
634	630, 633	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
635		ovex 7451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 + 1) ∈ V
636		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
637	636	anbi2d 642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
638		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = (𝑖 + 1) → (𝐶‘𝑘) = (𝐶‘(𝑖 + 1)))
639	638	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
640	637, 639	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
641	634, 635, 640, 228	vtoclf 3526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
642	627, 629, 641	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
643	642	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
644		nfv 1947	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
645		nfcv 2923	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑘0
646	459, 645	nffv 6893	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑘(𝐷‘0)
647	646, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝐷‘0):𝑋⟶ℂ
648	644, 647	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
649		c0ex 11293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
650		eleq1 2849	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
651	650	anbi2d 642	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
652		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 0 → (𝐷‘𝑘) = (𝐷‘0))
653	652	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
654	651, 653	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
655	648, 649, 654, 582	vtoclf 3526	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
656	1, 112, 655	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐷‘0):𝑋⟶ℂ)
657	656	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
658	657	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
659	643, 658	mulcld 11322	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
660	659	mullidd 11320	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
661	626, 660	eqtrd 2796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
662		1m1e0 12408	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 − 1) = 0
663	662	fveq2i 6886	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
664	3	eqcomi 2770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℤ≥‘0) = ℕ0
665	663, 664	eqtr2i 2785	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
666	665	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ≥‘(1 − 1)))
667	185, 666	eleqtrd 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘(1 − 1)))
668		fzdifsuc2 46295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (ℤ≥‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
669	667, 668	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
670	669	eqcomd 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
671	670	sumeq1d 15860	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
672	671	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
673	661, 672	oveq12d 7436	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
674	513, 613, 673	3eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
675	491, 495, 674	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
676		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑖C0)
677	345, 645	nffv 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝐶‘0)
678	677, 456	nffv 6893	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝐶‘0)‘𝑥)
679		nfcv 2923	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝑖 + 1) − 0)
680	459, 679	nffv 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(𝐷‘((𝑖 + 1) − 0))
681	680, 456	nffv 6893	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
682	678, 453, 681	nfov 7448	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
683	676, 453, 682	nfov 7448	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
684	664	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → (ℤ≥‘0) = ℕ0)
685	185, 684	eleqtrrd 2864	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘0))
686		eluzfz1 13657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑖))
687	685, 686	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
688	687	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑖))
689		oveq2 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
690	104	fveq1d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
691		oveq2 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
692	691	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
693	692	fveq1d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
694	690, 693	oveq12d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
695	689, 694	oveq12d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
696	470, 683, 439, 441, 688, 695	fsumsplit1 15904	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
697	620	subid1d 11651	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
698	697	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
699	698	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
700	699	oveq2d 7434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
701	700	oveq2d 7434	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
702	701	oveq1d 7433	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
703	702	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
704		bcn0 14447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
705	185, 704	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
706	705	oveq1d 7433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
707	706	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
708	677, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝐶‘0):𝑋⟶ℂ
709	644, 708	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
710	104	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
711	651, 710	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
712	709, 649, 711, 228	vtoclf 3526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
713	1, 112, 712	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐶‘0):𝑋⟶ℂ)
714	713	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
715	714	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
716	459, 631	nffv 6893	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝐷‘(𝑖 + 1))
717	716, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
718	630, 717	nfim 1929	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
719		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = (𝑖 + 1) → (𝐷‘𝑘) = (𝐷‘(𝑖 + 1)))
720	719	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
721	637, 720	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
722	718, 635, 721, 582	vtoclf 3526	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
723	627, 629, 722	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
724	723	ffvelcdmda 7082	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
725	715, 724	mulcld 11322	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
726	725	mullidd 11320	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
727	707, 726	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
728		nfv 1947	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗 𝑖 ∈ (0..^𝑁)
729		1zzd 12720	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
730	233	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
731		eldifi 4078	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
732		elfzelz 13649	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
733	731, 732	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
734	733	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
735		elfznn0 13747	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
736	731, 735	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
737		eldifsni 4753	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
738	736, 737	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0))
739		elnnne0 12613	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0))
740	738, 739	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
741		nnge1 12359	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ ℕ → 1 ≤ 𝑗)
742	740, 741	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
743	742	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
744		elfzle2 13654	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
745	731, 744	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≤ 𝑖)
746	745	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ≤ 𝑖)
747	729, 730, 734, 743, 746	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
748	747	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
749		0zd 12698	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
750		elfzel2 13647	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
751		elfzelz 13649	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
752	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
753	751	zred 12796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
754		1red 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
755	531	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...𝑖) → 0 < 1)
756		elfzle1 13653	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
757	752, 754, 753, 755, 756	ltletrd 11463	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
758	752, 753, 757	ltled 11451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
759		elfzle2 13654	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≤ 𝑖)
760	749, 750, 751, 758, 759	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
761	752, 757	gtned 11438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
762		nelsn 4627	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
763	761, 762	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
764	760, 763	eldifd 3910	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
765	764	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
766	765	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
767	748, 766	impbid 215	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
768	728, 767	alrimi 2250	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
769		dfcleq 2754	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
770	768, 769	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
771	770	sumeq1d 15860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
772	771	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
773	727, 772	oveq12d 7436	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
774	696, 703, 773	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
775	675, 774	oveq12d 7436	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
776		fzfid 14109	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) ∈ Fin)
777	185	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
778	765, 733	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
779		1zzd 12720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
780	778, 779	zsubcld 12801	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
781	777, 780	bccld 46299	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
782	781	nn0cnd 12662	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
783	782	adantll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
784	783	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
785		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋))
786		fzelp1 13703	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
787	786	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
788	785, 787, 551	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ)
789	787, 586	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
790	788, 789	mulcld 11322	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
791	784, 790	mulcld 11322	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
792	776, 791	fsumcl 15892	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
793	185	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
794		elfzelz 13649	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
795	794	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
796	793, 795	bccld 46299	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
797	796	nn0cnd 12662	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
798	797	adantll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
799	798	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
800		simpll 779	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑 ∧ 𝑖 ∈ (0..^𝑁)))
801		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥 ∈ 𝑋)
802	760	ssriv 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑖) ⊆ (0...𝑖)
803		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
804	802, 803	sselid 3929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
805	804	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
806	800, 801, 805, 433	syl21anc 851	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
807	805, 435	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
808	806, 807	mulcld 11322	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
809	799, 808	mulcld 11322	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
810	776, 809	fsumcl 15892	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
811	659, 792, 725, 810	add4d 11532	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
812		oveq1 7425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
813	812	oveq2d 7434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
814		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑘 → (𝐶‘𝑗) = (𝐶‘𝑘))
815	814	fveq1d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑘 → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘𝑘)‘𝑥))
816		oveq2 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
817	816	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
818	817	fveq1d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
819	815, 818	oveq12d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑘 → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
820	813, 819	oveq12d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
821		nfcv 2923	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝑖C(𝑗 − 1))
822	347, 456	nffv 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘((𝐶‘𝑗)‘𝑥)
823	569, 456	nffv 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
824	822, 453, 823	nfov 7448	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
825	821, 453, 824	nfov 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
826		nfcv 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
827	820, 825, 826	cbvsum 15855	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
828	827	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
829	828	oveq1d 7433	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
830		peano2zm 12732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
831	795, 830	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
832	793, 831	bccld 46299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
833	832	nn0cnd 12662	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
834	833	adantll 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
835	834	adantlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
836	835, 808	mulcld 11322	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
837	776, 836, 809	fsumadd 15899	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
838	837	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
839	833, 797	addcomd 11505	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
840		bcpasc 14458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
841	793, 795, 840	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
842	839, 841	eqtr2d 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
843	842	oveq1d 7433	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
844	843	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
845	844	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
846	835, 799, 808	adddird 11327	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
847	845, 846	eqtr2d 2797	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
848	847	sumeq2dv 15862	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
849	829, 838, 848	3eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
850	849	oveq2d 7434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
851		peano2nn0 12639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
852	793, 851	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
853	852, 795	bccld 46299	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
854	853	nn0cnd 12662	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
855	854	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
856	855	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
857	856, 808	mulcld 11322	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
858	776, 857	fsumcl 15892	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
859	659, 725, 858	addassd 11324	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
860	185, 851	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
861		bcn0 14447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
862	860, 861	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
863	862, 700	oveq12d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
864	863	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
865	864, 726	eqtr2d 2797	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
866	770	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
867	866	eqcomd 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
868	867	sumeq1d 15860	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
869	865, 868	oveq12d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
870		nfcv 2923	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘((𝑖 + 1)C0)
871	870, 453, 682	nfov 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
872	197, 851	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
873	872, 199	bccld 46299	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
874	873	nn0cnd 12662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
875	874	adantll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
876	875	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
877	876, 436	mulcld 11322	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
878		oveq2 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
879	878, 694	oveq12d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
880	470, 871, 439, 877, 688, 879	fsumsplit1 15904	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
881	880	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
882	869, 881	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
883	882	oveq2d 7434	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
884		bcnn 14449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
885	860, 884	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
886	885	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
887	886	oveq1d 7433	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
888	622	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
889	888	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
890	657, 889	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
891	890	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
892		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
893	891, 892	ffvelcdmd 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
894	643, 893	mulcld 11322	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
895	894	mullidd 11320	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
896	624	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
897	887, 895, 896	3eqtrrd 2801	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
898		fzdifsuc 13711	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (ℤ≥‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
899	685, 898	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
900	899	sumeq1d 15860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
901	900	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
902	897, 901	oveq12d 7436	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
903		nfcv 2923	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑖 + 1)C(𝑖 + 1))
904	632, 456	nffv 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝐶‘(𝑖 + 1))‘𝑥)
905		nfcv 2923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘((𝑖 + 1) − (𝑖 + 1))
906	459, 905	nffv 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
907	906, 456	nffv 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
908	904, 453, 907	nfov 7448	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
909	903, 453, 908	nfov 7448	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
910		fzfid 14109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...(𝑖 + 1)) ∈ Fin)
911	860	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
912		elfzelz 13649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
913	912	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
914	911, 913	bccld 46299	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
915	914	nn0cnd 12662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
916	915	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
917	916	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
918	627	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
919	90	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
920	206	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
921		elfzle1 13653	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
922	921	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
923	913	zred 12796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
924	911	nn0red 12661	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
925	211	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
926		elfzle2 13654	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
927	926	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
928	301	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
929	923, 924, 925, 927, 928	letrd 11460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ 𝑁)
930	919, 920, 913, 922, 929	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
931	930	adantll 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
932	918, 931, 228	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ)
933	932	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ)
934		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥 ∈ 𝑋)
935	933, 934	ffvelcdmd 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
936	918	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
937	590	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
938	937, 913	zsubcld 12801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
939	924, 923	subge0d 11899	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
940	927, 939	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘))
941	924, 923	resubcld 11737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
942	925, 923	resubcld 11737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ∈ ℝ)
943	925, 171, 246	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ)
944	924, 925, 923, 928	lesub1dd 11925	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘))
945	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℝ)
946	945, 923, 925, 922	lesub2dd 11926	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ≤ (𝑁 − 0))
947	941, 942, 943, 944, 946	letrd 11460	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0))
948	252	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁)
949	947, 948	breqtrd 5131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
950	919, 920, 938, 940, 949	elfzd 13640	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
951	950	adantll 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
952	951	adantlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
953		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷‘𝑗) = (𝐷‘((𝑖 + 1) − 𝑘)))
954	953	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷‘𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
955	310, 954	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
956	459, 346	nffv 6893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑘(𝐷‘𝑗)
957	956, 348, 349	nff 6703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑘(𝐷‘𝑗):𝑋⟶ℂ
958	343, 957	nfim 1929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)
959		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 𝑗 → (𝐷‘𝑘) = (𝐷‘𝑗))
960	959	feq1d 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑗 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘𝑗):𝑋⟶ℂ))
961	266, 960	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)))
962	958, 961, 582	chvarfv 2277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)
963	308, 955, 962	vtocl 3521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
964	936, 952, 963	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
965	964, 934	ffvelcdmd 7083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
966	935, 965	mulcld 11322	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
967	917, 966	mulcld 11322	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
968	860, 684	eleqtrrd 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ≥‘0))
969		eluzfz2 13658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 + 1) ∈ (ℤ≥‘0) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
970	968, 969	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
971	970	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
972		oveq2 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C(𝑖 + 1)))
973	638	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
974		oveq2 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = (𝑖 + 1) → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − (𝑖 + 1)))
975	974	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
976	975	fveq1d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
977	973, 976	oveq12d 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑖 + 1) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
978	972, 977	oveq12d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑖 + 1) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
979	470, 909, 910, 967, 971, 978	fsumsplit1 15904	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
980	979	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
981	883, 902, 980	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
982	850, 859, 981	3eqtrd 2800	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
983	775, 811, 982	3eqtrd 2800	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
984	438, 442, 983	3eqtrd 2800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
985	984	mpteq2dva 5198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
986	422, 985	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
987	986	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
988	189, 191, 987	3eqtrd 2800	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
989	178, 179, 182, 988	syl21anc 851	. . . . . 6 ⊢ ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
990	989	3exp 1137	. . . . 5 ⊢ (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
991	34, 47, 60, 73, 177, 990	fzind2 13916	. . . 4 ⊢ (𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))
992	21, 991	vtoclg 3518	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))
993	2, 6, 992	sylc 66	. 2 ⊢ (𝜑 → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))
994	1, 993	mpd 16	1 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  {cpr 4586   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ↑_pm cpm 8841  Fincfn 8966  ℂcc 11191  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198   < clt 11336   ≤ cle 11337   − cmin 11534  ℕcn 12328  ℕ₀cn0 12599  ℤcz 12686  ℤ_≥cuz 12958  ℝ⁺crp 13113  ...cfz 13632  ..^cfzo 13781  Ccbc 14439  Σcsu 15846   ↾_t crest 17584  TopOpenctopn 17585  ℂ_fldccnfld 21671   D cdv 26176   D^𝑛 cdvn 26177

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271  ax-addf 11272

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-fi 9396  df-sup 9427  df-inf 9428  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-q 13069  df-rp 13114  df-xneg 13234  df-xadd 13235  df-xmul 13236  df-icc 13476  df-fz 13633  df-fzo 13782  df-seq 14138  df-exp 14198  df-fac 14411  df-bc 14440  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-sum 15847  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-starv 17436  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-unif 17444  df-hom 17445  df-cco 17446  df-rest 17586  df-topn 17587  df-0g 17605  df-gsum 17606  df-topgen 17607  df-pt 17608  df-prds 17611  df-xrs 17667  df-qtop 17672  df-imas 17673  df-xps 17675  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-submnd 18972  df-mulg 19271  df-cntz 19524  df-cmn 19989  df-psmet 21663  df-xmet 21664  df-met 21665  df-bl 21666  df-mopn 21667  df-fbas 21668  df-fg 21669  df-cnfld 21672  df-top 23205  df-topon 23222  df-topsp 23244  df-bases 23257  df-cld 23330  df-ntr 23331  df-cls 23332  df-nei 23409  df-lp 23447  df-perf 23448  df-cn 23538  df-cnp 23539  df-haus 23626  df-tx 23874  df-hmeo 24067  df-fil 24158  df-fm 24250  df-flim 24251  df-flf 24252  df-xms 24632  df-ms 24633  df-tms 24634  df-cncf 25192  df-limc 26179  df-dv 26180  df-dvn 26181

This theorem is used by:  dvnprodlem2  46926