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Theorem dvnmul 42451
Description: Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
dvnmul.s (𝜑𝑆 ∈ {ℝ, ℂ})
dvnmul.x (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
dvnmul.a ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
dvnmul.cc ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)
dvnmul.n (𝜑𝑁 ∈ ℕ0)
dvnmulf 𝐹 = (𝑥𝑋𝐴)
dvnmul.f 𝐺 = (𝑥𝑋𝐵)
dvnmul.dvnf ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
dvnmul.dvng ((𝜑𝑘 ∈ (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 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘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 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 dvnmul.n . . 3 (𝜑𝑁 ∈ ℕ0)
3 nn0uz 12275 . . . . 5 0 = (ℤ‘0)
42, 3eleqtrdi 2926 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12917 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
7 eleq1 2903 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8 fveq2 6659 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9 oveq2 7154 . . . . . . . . . 10 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
109sumeq1d 15056 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
11 oveq1 7153 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12 fvoveq1 7169 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝐷‘(𝑛𝑘)) = (𝐷‘(𝑁𝑘)))
1312fveq1d 6661 . . . . . . . . . . . 12 (𝑛 = 𝑁 → ((𝐷‘(𝑛𝑘))‘𝑥) = ((𝐷‘(𝑁𝑘))‘𝑥))
1413oveq2d 7162 . . . . . . . . . . 11 (𝑛 = 𝑁 → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))
1511, 14oveq12d 7164 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1615sumeq2sdv 15059 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1710, 16eqtrd 2859 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1817mpteq2dv 5149 . . . . . . 7 (𝑛 = 𝑁 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))
198, 18eqeq12d 2840 . . . . . 6 (𝑛 = 𝑁 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))))
2019imbi2d 344 . . . . 5 (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))))
217, 20imbi12d 348 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))))) ↔ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))))))
22 fveq2 6659 . . . . . . 7 (𝑚 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0))
23 simpl 486 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑥𝑋) → 𝑚 = 0)
2423oveq2d 7162 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝑥𝑋) → (0...𝑚) = (0...0))
25 simpll 766 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
2625oveq1d 7161 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
2725fvoveq1d 7168 . . . . . . . . . . . 12 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(0 − 𝑘)))
2827fveq1d 6661 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
2928oveq2d 7162 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
3026, 29oveq12d 7164 . . . . . . . . 9 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3124, 30sumeq12rdv 15062 . . . . . . . 8 ((𝑚 = 0 ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3231mpteq2dva 5148 . . . . . . 7 (𝑚 = 0 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
3322, 32eqeq12d 2840 . . . . . 6 (𝑚 = 0 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
3433imbi2d 344 . . . . 5 (𝑚 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))))
35 fveq2 6659 . . . . . . 7 (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
36 simpl 486 . . . . . . . . . 10 ((𝑚 = 𝑖𝑥𝑋) → 𝑚 = 𝑖)
3736oveq2d 7162 . . . . . . . . 9 ((𝑚 = 𝑖𝑥𝑋) → (0...𝑚) = (0...𝑖))
38 simpll 766 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
3938oveq1d 7161 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
4038fvoveq1d 7168 . . . . . . . . . . . 12 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑖𝑘)))
4140fveq1d 6661 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑖𝑘))‘𝑥))
4241oveq2d 7162 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
4339, 42oveq12d 7164 . . . . . . . . 9 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4437, 43sumeq12rdv 15062 . . . . . . . 8 ((𝑚 = 𝑖𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4544mpteq2dva 5148 . . . . . . 7 (𝑚 = 𝑖 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))
4635, 45eqeq12d 2840 . . . . . 6 (𝑚 = 𝑖 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))))
4746imbi2d 344 . . . . 5 (𝑚 = 𝑖 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))))
48 fveq2 6659 . . . . . . 7 (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
49 simpl 486 . . . . . . . . . 10 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → 𝑚 = (𝑖 + 1))
5049oveq2d 7162 . . . . . . . . 9 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
51 simpll 766 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
5251oveq1d 7161 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
5351fvoveq1d 7168 . . . . . . . . . . . 12 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
5453fveq1d 6661 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
5554oveq2d 7162 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
5652, 55oveq12d 7164 . . . . . . . . 9 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
5750, 56sumeq12rdv 15062 . . . . . . . 8 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
5857mpteq2dva 5148 . . . . . . 7 (𝑚 = (𝑖 + 1) → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
5948, 58eqeq12d 2840 . . . . . 6 (𝑚 = (𝑖 + 1) → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
6059imbi2d 344 . . . . 5 (𝑚 = (𝑖 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
61 fveq2 6659 . . . . . . 7 (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
62 simpl 486 . . . . . . . . . 10 ((𝑚 = 𝑛𝑥𝑋) → 𝑚 = 𝑛)
6362oveq2d 7162 . . . . . . . . 9 ((𝑚 = 𝑛𝑥𝑋) → (0...𝑚) = (0...𝑛))
64 simpll 766 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
6564oveq1d 7161 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
6664fvoveq1d 7168 . . . . . . . . . . . 12 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑛𝑘)))
6766fveq1d 6661 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑛𝑘))‘𝑥))
6867oveq2d 7162 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))
6965, 68oveq12d 7164 . . . . . . . . 9 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7063, 69sumeq12rdv 15062 . . . . . . . 8 ((𝑚 = 𝑛𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7170mpteq2dva 5148 . . . . . . 7 (𝑚 = 𝑛 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))))
7261, 71eqeq12d 2840 . . . . . 6 (𝑚 = 𝑛 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))))
7372imbi2d 344 . . . . 5 (𝑚 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))))))
74 dvnmul.s . . . . . . . . 9 (𝜑𝑆 ∈ {ℝ, ℂ})
75 recnprss 24505 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
7674, 75syl 17 . . . . . . . 8 (𝜑𝑆 ⊆ ℂ)
77 dvnmul.a . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
78 dvnmul.cc . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)
7977, 78mulcld 10655 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) ∈ ℂ)
80 restsspw 16703 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
81 dvnmul.x . . . . . . . . . . 11 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
8280, 81sseldi 3951 . . . . . . . . . 10 (𝜑𝑋 ∈ 𝒫 𝑆)
83 elpwi 4531 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑆𝑋𝑆)
8482, 83syl 17 . . . . . . . . 9 (𝜑𝑋𝑆)
85 cnex 10612 . . . . . . . . . 10 ℂ ∈ V
8685a1i 11 . . . . . . . . 9 (𝜑 → ℂ ∈ V)
8779, 84, 86, 74mptelpm 41663 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
88 dvn0 24525 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
8976, 87, 88syl2anc 587 . . . . . . 7 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
90 0z 11987 . . . . . . . . . . . 12 0 ∈ ℤ
91 fzsn 12951 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
9290, 91ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
9392sumeq1i 15053 . . . . . . . . . 10 Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
9493a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
95 nfcvd 2983 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑘(𝐴 · 𝐵))
96 nfv 1916 . . . . . . . . . 10 𝑘(𝜑𝑥𝑋)
97 oveq2 7154 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C𝑘) = (0C0))
98 0nn0 11907 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
99 bcn0 13673 . . . . . . . . . . . . . . . 16 (0 ∈ ℕ0 → (0C0) = 1)
10098, 99ax-mp 5 . . . . . . . . . . . . . . 15 (0C0) = 1
101100a1i 11 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C0) = 1)
10297, 101eqtrd 2859 . . . . . . . . . . . . 13 (𝑘 = 0 → (0C𝑘) = 1)
103102adantl 485 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
104 fveq2 6659 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐶𝑘) = (𝐶‘0))
105104adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶𝑘) = (𝐶‘0))
106 dvnmul.c . . . . . . . . . . . . . . . . . . . . 21 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
107 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
108107cbvmptv 5156 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
109106, 108eqtri 2847 . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
110 fveq2 6659 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
111 eluzfz1 12916 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
1124, 111syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 ∈ (0...𝑁))
113 fvexd 6674 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
114109, 110, 112, 113fvmptd3 6780 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
115114adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
116105, 115eqtrd 2859 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘0))
117 dvnmulf . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑥𝑋𝐴)
11877, 84, 86, 74mptelpm 41663 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
119117, 118eqeltrid 2920 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
120 dvn0 24525 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
12176, 119, 120syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
122121adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
123116, 122eqtrd 2859 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 0) → (𝐶𝑘) = 𝐹)
124123fveq1d 6661 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
125124adantlr 714 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
126 simpr 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
127117fvmpt2 6768 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝐴 ∈ ℂ) → (𝐹𝑥) = 𝐴)
128126, 77, 127syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐹𝑥) = 𝐴)
129128adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐹𝑥) = 𝐴)
130125, 129eqtrd 2859 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = 𝐴)
131 oveq2 7154 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
132 0m0e0 11752 . . . . . . . . . . . . . . . . . . . 20 (0 − 0) = 0
133132a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 0) = 0)
134131, 133eqtrd 2859 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (0 − 𝑘) = 0)
135134fveq2d 6663 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
136135fveq1d 6661 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
137136adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
138137adantlr 714 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
139 dvnmul.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
140 fveq2 6659 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
141140cbvmptv 5156 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
142139, 141eqtri 2847 . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
143142fveq1i 6660 . . . . . . . . . . . . . . . . . 18 (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)
144143a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0))
145 eqidd 2825 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
146 fveq2 6659 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
147146adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
148 dvnmul.f . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑋𝐵)
14978, 84, 86, 74mptelpm 41663 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝑋𝐵) ∈ (ℂ ↑pm 𝑆))
150148, 149eqeltrid 2920 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐺 ∈ (ℂ ↑pm 𝑆))
151 dvn0 24525 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
15276, 150, 151syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
153152adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
154147, 153eqtrd 2859 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
155148a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 = (𝑥𝑋𝐵))
156 mptexg 6973 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ 𝒫 𝑆 → (𝑥𝑋𝐵) ∈ V)
15782, 156syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝑋𝐵) ∈ V)
158155, 157eqeltrd 2916 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ V)
159145, 154, 112, 158fvmptd 6764 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
160144, 159eqtrd 2859 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐷‘0) = 𝐺)
161160fveq1d 6661 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
162161ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
163155, 78fvmpt2d 6770 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐺𝑥) = 𝐵)
164163adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐺𝑥) = 𝐵)
165138, 162, 1643eqtrd 2863 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
166130, 165oveq12d 7164 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
167103, 166oveq12d 7164 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
16879mulid2d 10653 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
169168adantr 484 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
170167, 169eqtrd 2859 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
171 0re 10637 . . . . . . . . . . 11 0 ∈ ℝ
172171a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 ∈ ℝ)
17395, 96, 170, 172, 79sumsnd 41515 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
17494, 173eqtr2d 2860 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
175174mpteq2dva 5148 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
17689, 175eqtrd 2859 . . . . . 6 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
177176a1i 11 . . . . 5 (𝑁 ∈ (ℤ‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
178 simp3 1135 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑)
179 simp1 1133 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → 𝑖 ∈ (0..^𝑁))
180 simp2 1134 . . . . . . . 8 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))))
181 pm3.35 802 . . . . . . . 8 ((𝜑 ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))
182178, 180, 181syl2anc 587 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))
18376adantr 484 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
18487adantr 484 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
185 elfzonn0 13084 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
186185adantl 485 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
187 dvnp1 24526 . . . . . . . . . 10 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑖 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
188183, 184, 186, 187syl3anc 1368 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
189188adantr 484 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
190 simpr 488 . . . . . . . . 9 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))
191190oveq2d 7162 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) = (𝑆 D (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))))
192 eqid 2824 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
193 eqid 2824 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
19474adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19581adantr 484 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
196 fzfid 13343 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
197185adantr 484 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
198 elfzelz 12909 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
199198adantl 485 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
200197, 199bccld 41813 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
201200nn0cnd 11952 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
202201adantll 713 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
2032023adant3 1129 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
204 simpll 766 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
205 0zd 11988 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
206 elfzoel2 13039 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
207206adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
208205, 207, 1993jca 1125 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
209 elfzle1 12912 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
210209adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
211199zred 12082 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
212206zred 12082 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
213212adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
214185nn0red 11951 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
215214adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
216 elfzle2 12913 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑘𝑖)
217216adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑖)
218 elfzolt2 13049 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
219218adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
220211, 215, 213, 217, 219lelttrd 10792 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
221211, 213, 220ltled 10782 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑁)
222208, 210, 221jca32 519 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
223 elfz2 12899 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
224222, 223sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
225224adantll 713 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
226 dvnmul.dvnf . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
227106a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
228 fvexd 6674 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
229227, 228fvmpt2d 6770 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
230229feq1d 6488 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐶𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
231226, 230mpbird 260 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ)
232204, 225, 231syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘):𝑋⟶ℂ)
2332323adant3 1129 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐶𝑘):𝑋⟶ℂ)
234 simp3 1135 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → 𝑥𝑋)
235233, 234ffvelrnd 6841 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
236185nn0zd 12080 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
237236adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
238237, 199zsubcld 12087 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℤ)
239205, 207, 2383jca 1125 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ))
240 elfzel2 12907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
241240zred 12082 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
242198zred 12082 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
243241, 242subge0d 11224 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖𝑘) ↔ 𝑘𝑖))
244216, 243mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖𝑘))
245244adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖𝑘))
246215, 211resubcld 11062 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℝ)
247213, 211resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ∈ ℝ)
248171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
249213, 248jca 515 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
250 resubcl 10944 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
251249, 250syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
252215, 213, 211, 219ltsub1dd 11246 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁𝑘))
253248, 211, 213, 210lesub2dd 11251 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ (𝑁 − 0))
254246, 247, 251, 252, 253ltletrd 10794 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁 − 0))
255212recnd 10663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
256255subid1d 10980 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
257256adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
258254, 257breqtrd 5079 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < 𝑁)
259246, 213, 258ltled 10782 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ≤ 𝑁)
260239, 245, 259jca32 519 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
261 elfz2 12899 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
262260, 261sylibr 237 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
263262adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
264 ovex 7179 . . . . . . . . . . . . . . . . . 18 (𝑖𝑘) ∈ V
265 eleq1 2903 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖𝑘) ∈ (0...𝑁)))
266265anbi2d 631 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁))))
267 fveq2 6659 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
268267feq1d 6488 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
269266, 268imbi12d 348 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)))
270 nfv 1916 . . . . . . . . . . . . . . . . . . 19 𝑘((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
271 eleq1 2903 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
272271anbi2d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑𝑗 ∈ (0...𝑁))))
273 fveq2 6659 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
274273feq1d 6488 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
275272, 274imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
276 dvnmul.dvng . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
277270, 275, 276chvarfv 2244 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
278264, 269, 277vtocl 3545 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
279204, 263, 278syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
280 fveq2 6659 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
281 fvexd 6674 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) ∈ V)
282142, 280, 262, 281fvmptd3 6780 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
283282adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
284283feq1d 6488 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
285279, 284mpbird 260 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
2862853adant3 1129 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
287286, 234ffvelrnd 6841 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
288235, 287mulcld 10655 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
289203, 288mulcld 10655 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
2902033expa 1115 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
291237peano2zd 12085 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
292291, 199zsubcld 12087 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
293205, 207, 2923jca 1125 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
294 peano2re 10807 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
295241, 294syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
296 peano2re 10807 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
297242, 296syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
298242ltp1d 11564 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
299 1red 10636 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
300242, 241, 299, 216leadd1dd 11248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
301242, 297, 295, 298, 300ltletrd 10794 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
302242, 295, 301ltled 10782 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
303302adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
304215, 294syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
305304, 211subge0d 11224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
306303, 305mpbird 260 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
307304, 211resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
308 elfzop1le2 41787 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
309308adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
310304, 213, 211, 309lesub1dd 11250 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
311253, 257breqtrd 5079 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ 𝑁)
312307, 247, 213, 310, 311letrd 10791 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
313293, 306, 312jca32 519 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
314 elfz2 12899 . . . . . . . . . . . . . . . . . . . . 21 (((𝑖 + 1) − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
315313, 314sylibr 237 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
316315adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
317 ovex 7179 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) − 𝑘) ∈ V
318 eleq1 2903 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
319318anbi2d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
320 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
321320feq1d 6488 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
322319, 321imbi12d 348 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
323317, 322, 277vtocl 3545 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
324204, 316, 323syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
325142a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
326 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
327326fveq2d 6663 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
328 fvexd 6674 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
329325, 327, 316, 328fvmptd 6764 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
330329feq1d 6488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
331324, 330mpbird 260 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
332331ffvelrnda 6840 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
3332353expa 1115 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
334332, 333mulcomd 10656 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
335334oveq2d 7162 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) = ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
336199peano2zd 12085 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
337205, 207, 3363jca 1125 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ))
338171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
339338, 242, 297, 209, 298lelttrd 10792 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
340338, 297, 339ltled 10782 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
341340adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
342211, 296syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
343300adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
344342, 304, 213, 343, 309letrd 10791 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
345337, 341, 344jca32 519 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
346 elfz2 12899 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 + 1) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
347345, 346sylibr 237 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
348347adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
349 ovex 7179 . . . . . . . . . . . . . . . . . . 19 (𝑘 + 1) ∈ V
350 eleq1 2903 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
351350anbi2d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
352 fveq2 6659 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝐶𝑗) = (𝐶‘(𝑘 + 1)))
353352feq1d 6488 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝐶𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
354351, 353imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 1) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
355 nfv 1916 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝜑𝑗 ∈ (0...𝑁))
356 nfmpt1 5151 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
357106, 356nfcxfr 2980 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝐶
358 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑗
359357, 358nffv 6669 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝐶𝑗)
360 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑋
361 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . 22 𝑘
362359, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶𝑗):𝑋⟶ℂ
363355, 362nfim 1898 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
364 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑗 → (𝐶𝑘) = (𝐶𝑗))
365364feq1d 6488 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶𝑗):𝑋⟶ℂ))
366272, 365imbi12d 348 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)))
367363, 366, 231chvarfv 2244 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
368349, 354, 367vtocl 3545 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
369204, 348, 368syl2anc 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
370369ffvelrnda 6840 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
3712873expa 1115 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
372370, 371mulcld 10655 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
373332, 333mulcld 10655 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) ∈ ℂ)
374372, 373addcld 10654 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) ∈ ℂ)
375335, 374eqeltrrd 2917 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
376290, 375mulcld 10655 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
3773763impa 1107 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
378204, 74syl 17 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
379171a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → 0 ∈ ℝ)
380204, 81syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
381378, 380, 202dvmptconst 42423 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ (𝑖C𝑘))) = (𝑥𝑋 ↦ 0))
3822883expa 1115 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
383204, 225, 229syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
384383eqcomd 2830 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶𝑘))
385232feqmptd 6722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)))
386384, 385eqtr2d 2860 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
387386oveq2d 7162 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
388378, 75syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
389204, 119syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
390 elfznn0 13002 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
391390adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
392 dvnp1 24526 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
393388, 389, 391, 392syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
394393eqcomd 2830 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
395 fveq2 6659 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
396 fvexd 6674 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
397109, 395, 348, 396fvmptd3 6780 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
398397eqcomd 2830 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
399369feqmptd 6722 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
400398, 399eqtrd 2859 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
401387, 394, 4003eqtrd 2863 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
402283eqcomd 2830 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) = (𝐷‘(𝑖𝑘)))
403285feqmptd 6722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)))
404402, 403eqtr2d 2860 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
405404oveq2d 7162 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
406204, 150syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
407 fznn0sub 12941 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → (𝑖𝑘) ∈ ℕ0)
408407adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℕ0)
409 dvnp1 24526 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
410388, 406, 408, 409syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
411410eqcomd 2830 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)))
412215recnd 10663 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
413 1cnd 10630 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
414211recnd 10663 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
415412, 413, 414addsubd 11012 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖𝑘) + 1))
416415eqcomd 2830 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖𝑘) + 1) = ((𝑖 + 1) − 𝑘))
417416fveq2d 6663 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
418417adantll 713 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
419329eqcomd 2830 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
420331feqmptd 6722 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
421418, 419, 4203eqtrd 2863 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
422405, 411, 4213eqtrd 2863 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
423378, 333, 370, 401, 371, 332, 422dvmptmul 24562 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))) = (𝑥𝑋 ↦ ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)))))
424378, 290, 379, 381, 382, 374, 423dvmptmul 24562 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) = (𝑥𝑋 ↦ ((0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘)))))
425382mul02d 10832 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = 0)
426335oveq1d 7161 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘)) = (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)))
427375, 290mulcomd 10656 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
428426, 427eqtrd 2859 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
429425, 428oveq12d 7164 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘))) = (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
430376addid2d 10835 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
431429, 430eqtrd 2859 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
432431mpteq2dva 5148 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) · (𝑖C𝑘)))) = (𝑥𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
433424, 432eqtrd 2859 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) = (𝑥𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
434192, 193, 194, 195, 196, 289, 377, 433dvmptfsum 24576 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
435202adantlr 714 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
436372an32s 651 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
437 anass 472 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)))
438 ancom 464 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) ↔ (𝑥𝑋𝑘 ∈ (0...𝑖)))
439438anbi2i 625 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
440 anass 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
441440bicomi 227 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
442439, 441bitri 278 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
443437, 442bitri 278 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
444443imbi1i 353 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ))
445333, 444mpbi 233 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
446443imbi1i 353 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ))
447332, 446mpbi 233 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
448445, 447mulcld 10655 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
449435, 436, 448adddid 10659 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
450449sumeq2dv 15058 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
451196adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...𝑖) ∈ Fin)
452435, 436mulcld 10655 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
453435, 448mulcld 10655 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
454451, 452, 453fsumadd 15094 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
455 oveq2 7154 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (𝑖C𝑘) = (𝑖C))
456 fvoveq1 7169 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐶‘(𝑘 + 1)) = (𝐶‘( + 1)))
457456fveq1d 6661 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘( + 1))‘𝑥))
458 oveq2 7154 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑖𝑘) = (𝑖))
459458fveq2d 6663 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐷‘(𝑖𝑘)) = (𝐷‘(𝑖)))
460459fveq1d 6661 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐷‘(𝑖𝑘))‘𝑥) = ((𝐷‘(𝑖))‘𝑥))
461457, 460oveq12d 7164 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) = (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
462455, 461oveq12d 7164 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))))
463 nfcv 2982 . . . . . . . . . . . . . . . . . 18 (0...𝑖)
464 nfcv 2982 . . . . . . . . . . . . . . . . . 18 𝑘(0...𝑖)
465 nfcv 2982 . . . . . . . . . . . . . . . . . 18 ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
466 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑘(𝑖C)
467 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑘 ·
468 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . 22 𝑘( + 1)
469357, 468nffv 6669 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶‘( + 1))
470 nfcv 2982 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑥
471469, 470nffv 6669 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐶‘( + 1))‘𝑥)
472 nfmpt1 5151 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
473139, 472nfcxfr 2980 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐷
474 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝑖)
475473, 474nffv 6669 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐷‘(𝑖))
476475, 470nffv 6669 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐷‘(𝑖))‘𝑥)
477471, 467, 476nfov 7176 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))
478466, 467, 477nfov 7176 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
479462, 463, 464, 465, 478cbvsum 15050 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
480479a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))))
481 1zzd 12008 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 1 ∈ ℤ)
48290a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ ℤ)
483236ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑖 ∈ ℤ)
484 nfv 1916 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
485 nfcv 2982 . . . . . . . . . . . . . . . . . . . . 21 𝑘
486485, 464nfel 2996 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...𝑖)
487484, 486nfan 1901 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))
488478, 361nfel 2996 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ
489487, 488nfim 1898 . . . . . . . . . . . . . . . . . 18 𝑘((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
490 eleq1 2903 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 ∈ (0...𝑖) ↔ ∈ (0...𝑖)))
491490anbi2d 631 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))))
492462eleq1d 2900 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ))
493491, 492imbi12d 348 . . . . . . . . . . . . . . . . . 18 (𝑘 = → (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)))
494489, 493, 452chvarfv 2244 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
495 oveq2 7154 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (𝑖C) = (𝑖C(𝑗 − 1)))
496 fvoveq1 7169 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐶‘( + 1)) = (𝐶‘((𝑗 − 1) + 1)))
497496fveq1d 6661 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐶‘( + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
498 oveq2 7154 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → (𝑖) = (𝑖 − (𝑗 − 1)))
499498fveq2d 6663 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐷‘(𝑖)) = (𝐷‘(𝑖 − (𝑗 − 1))))
500499fveq1d 6661 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐷‘(𝑖))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
501497, 500oveq12d 7164 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
502495, 501oveq12d 7164 . . . . . . . . . . . . . . . . 17 ( = (𝑗 − 1) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
503481, 482, 483, 494, 502fsumshft 15133 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
504480, 503eqtrd 2859 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
505 0p1e1 11754 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
506505oveq1i 7156 . . . . . . . . . . . . . . . . 17 ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
507506sumeq1i 15053 . . . . . . . . . . . . . . . 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)))
508507a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 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))))
509 elfzelz 12909 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
510509zcnd 12083 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
511 1cnd 10630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
512510, 511npcand 10995 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
513512fveq2d 6663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶𝑗))
514513fveq1d 6661 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
515514adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
516214recnd 10663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
517516adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
518510adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
519511adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
520517, 518, 519subsub3d 11021 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
521520fveq2d 6663 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
522521fveq1d 6661 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
523515, 522oveq12d 7164 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) = (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
524523oveq2d 7162 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
525524sumeq2dv 15058 . . . . . . . . . . . . . . . . 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)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
526525ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
527 nfv 1916 . . . . . . . . . . . . . . . . 17 𝑗((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
528 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
529 fzfid 13343 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...(𝑖 + 1)) ∈ Fin)
530185adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
531509adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
532 1zzd 12008 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
533531, 532zsubcld 12087 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
534530, 533bccld 41813 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
535534nn0cnd 11952 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
536535adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
537536adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
5381ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
539 0zd 11988 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
540206adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
541539, 540, 5313jca 1125 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ))
542171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
543509zred 12082 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
544 1red 10636 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
545 0lt1 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 < 1
546545a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
547 elfzle1 12912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
548542, 544, 543, 546, 547ltletrd 10794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
549542, 543, 548ltled 10782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
550549adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
551543adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
552214adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
553 1red 10636 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
554552, 553readdcld 10664 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
555212adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
556 elfzle2 12913 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
557556adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
558308adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
559551, 554, 555, 557, 558letrd 10791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗𝑁)
560541, 550, 559jca32 519 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
561 elfz2 12899 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
562560, 561sylibr 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
563562adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
564538, 563, 367syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
565564adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
566 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥𝑋)
567565, 566ffvelrnd 6841 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
568236adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
569568peano2zd 12085 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
570569, 531zsubcld 12087 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
571539, 540, 5703jca 1125 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ))
572554, 551subge0d 11224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
573557, 572mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
574554, 551resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
575574leidd 11200 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
576543, 548elrpd 12423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
577576adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
578554, 577ltsubrpd 12458 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
579574, 554, 555, 578, 558ltletrd 10794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
580574, 574, 555, 575, 579lelttrd 10792 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
581574, 555, 580ltled 10782 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
582571, 573, 581jca32 519 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
583 elfz2 12899 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 + 1) − 𝑗) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
584582, 583sylibr 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
585584adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
586 nfv 1916 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
587 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘((𝑖 + 1) − 𝑗)
588473, 587nffv 6669 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘((𝑖 + 1) − 𝑗))
589588, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
590586, 589nfim 1898 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
591 ovex 7179 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 + 1) − 𝑗) ∈ V
592 eleq1 2903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
593592anbi2d 631 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
594 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
595594feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
596593, 595imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
597139a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
598 fvexd 6674 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
599597, 598fvmpt2d 6770 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
600599feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐷𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
601276, 600mpbird 260 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ)
602590, 591, 596, 601vtoclf 3544 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
603538, 585, 602syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
604603adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
605604, 566ffvelrnd 6841 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
606567, 605mulcld 10655 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
607537, 606mulcld 10655 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
608 1zzd 12008 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
609236peano2zd 12085 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
610505eqcomi 2833 . . . . . . . . . . . . . . . . . . . . . . 23 1 = (0 + 1)
611610a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
612171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
613 1red 10636 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
614185nn0ge0d 11953 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
615612, 214, 613, 614leadd1dd 11248 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
616611, 615eqbrtrd 5075 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
617608, 609, 6163jca 1125 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
618 eluz2 12244 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
619617, 618sylibr 237 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ‘1))
620 eluzfz2 12917 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
621619, 620syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
622621ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
623 oveq1 7153 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
624623oveq2d 7162 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
625 fveq2 6659 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐶𝑗) = (𝐶‘(𝑖 + 1)))
626625fveq1d 6661 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐶𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
627 oveq2 7154 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
628627fveq2d 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
629628fveq1d 6661 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
630626, 629oveq12d 7164 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
631624, 630oveq12d 7164 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑖 + 1) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
632527, 528, 529, 607, 622, 631fsumsplit1 42080 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘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)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
633 1cnd 10630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
634516, 633pncand 10992 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
635634oveq2d 7162 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
636 bcnn 13675 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
637185, 636syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
638635, 637eqtrd 2859 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
639516, 633addcld 10654 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
640639subidd 10979 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
641640fveq2d 6663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
642641fveq1d 6661 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
643642oveq2d 7162 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
644638, 643oveq12d 7164 . . . . . . . . . . . . . . . . . . 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))))
645644ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))))
646 simpl 486 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝜑)
647 fzofzp1 13136 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
648647adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
649 nfv 1916 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
650 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝑖 + 1)
651357, 650nffv 6669 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐶‘(𝑖 + 1))
652651, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
653649, 652nfim 1898 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
654 ovex 7179 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 + 1) ∈ V
655 eleq1 2903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
656655anbi2d 631 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
657 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝐶𝑘) = (𝐶‘(𝑖 + 1)))
658657feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
659656, 658imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
660653, 654, 659, 231vtoclf 3544 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
661646, 648, 660syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
662661ffvelrnda 6840 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
663 nfv 1916 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
664 nfcv 2982 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑘0
665473, 664nffv 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝐷‘0)
666665, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘0):𝑋⟶ℂ
667663, 666nfim 1898 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
668 c0ex 10629 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
669 eleq1 2903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
670669anbi2d 631 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
671 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝐷𝑘) = (𝐷‘0))
672671feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
673670, 672imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
674667, 668, 673, 601vtoclf 3544 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
6751, 112, 674syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐷‘0):𝑋⟶ℂ)
676675adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
677676ffvelrnda 6840 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
678662, 677mulcld 10655 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
679678mulid2d 10653 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
680645, 679eqtrd 2859 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
681 1m1e0 11704 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 − 1) = 0
682681fveq2i 6662 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘(1 − 1)) = (ℤ‘0)
6833eqcomi 2833 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘0) = ℕ0
684682, 683eqtr2i 2848 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘(1 − 1))
685684a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ‘(1 − 1)))
686185, 685eleqtrd 2918 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘(1 − 1)))
687 fzdifsuc2 41808 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (ℤ‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
688686, 687syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
689688eqcomd 2830 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
690689sumeq1d 15056 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
691690ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
692680, 691oveq12d 7164 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
693526, 632, 6923eqtrd 2863 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 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)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
694504, 508, 6933eqtrd 2863 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
695 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑘(𝑖C0)
696357, 664nffv 6669 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐶‘0)
697696, 470nffv 6669 . . . . . . . . . . . . . . . . . 18 𝑘((𝐶‘0)‘𝑥)
698 nfcv 2982 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1) − 0)
699473, 698nffv 6669 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐷‘((𝑖 + 1) − 0))
700699, 470nffv 6669 . . . . . . . . . . . . . . . . . 18 𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
701697, 467, 700nfov 7176 . . . . . . . . . . . . . . . . 17 𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
702695, 467, 701nfov 7176 . . . . . . . . . . . . . . . 16 𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
703683a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (ℤ‘0) = ℕ0)
704185, 703eleqtrrd 2919 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘0))
705 eluzfz1 12916 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ‘0) → 0 ∈ (0...𝑖))
706704, 705syl 17 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
707706ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ (0...𝑖))
708 oveq2 7154 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
709104fveq1d 6661 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐶𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
710 oveq2 7154 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
711710fveq2d 6663 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
712711fveq1d 6661 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
713709, 712oveq12d 7164 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
714708, 713oveq12d 7164 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
715484, 702, 451, 453, 707, 714fsumsplit1 42080 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
716639subid1d 10980 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
717716fveq2d 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
718717fveq1d 6661 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
719718oveq2d 7162 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
720719oveq2d 7162 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
721720oveq1d 7161 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑁) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
722721ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
723 bcn0 13673 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
724185, 723syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
725724oveq1d 7161 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
726725ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
727696, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘0):𝑋⟶ℂ
728663, 727nfim 1898 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
729104feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
730670, 729imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
731728, 668, 730, 231vtoclf 3544 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
7321, 112, 731syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐶‘0):𝑋⟶ℂ)
733732adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
734733ffvelrnda 6840 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
735473, 650nffv 6669 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘(𝑖 + 1))
736735, 360, 361nff 6499 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
737649, 736nfim 1898 . . . . . . . . . . . . . . . . . . . . . 22 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
738 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → (𝐷𝑘) = (𝐷‘(𝑖 + 1)))
739738feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
740656, 739imbi12d 348 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
741737, 654, 740, 601vtoclf 3544 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
742646, 648, 741syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
743742ffvelrnda 6840 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
744734, 743mulcld 10655 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
745744mulid2d 10653 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
746726, 745eqtrd 2859 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
747 nfv 1916 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑖 ∈ (0..^𝑁)
748 1zzd 12008 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
749236adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
750 eldifi 4089 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
751 elfzelz 12909 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
752750, 751syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
753752adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
754748, 749, 7533jca 1125 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → (1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
755 elfznn0 13002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
756750, 755syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
757 eldifsni 4707 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
758756, 757jca 515 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0𝑗 ≠ 0))
759 elnnne0 11906 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0𝑗 ≠ 0))
760758, 759sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
761 nnge1 11660 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ → 1 ≤ 𝑗)
762760, 761syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
763762adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
764 elfzle2 12913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
765750, 764syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗𝑖)
766765adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗𝑖)
767754, 763, 766jca32 519 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
768 elfz2 12899 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) ↔ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
769767, 768sylibr 237 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
770769ex 416 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
771 0zd 11988 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
772 elfzel2 12907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
773 elfzelz 12909 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
774771, 772, 7733jca 1125 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → (0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
775171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
776773zred 12082 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
777 1red 10636 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
778545a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 0 < 1)
779 elfzle1 12912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
780775, 777, 776, 778, 779ltletrd 10794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
781775, 776, 780ltled 10782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
782 elfzle2 12913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 𝑗𝑖)
783774, 781, 782jca32 519 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
784 elfz2 12899 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) ↔ ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
785783, 784sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
786775, 780gtned 10769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
787 nelsn 4590 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
788786, 787syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
789785, 788eldifd 3930 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
790789adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
791790ex 416 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
792770, 791impbid 215 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
793747, 792alrimi 2215 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
794 dfcleq 2818 . . . . . . . . . . . . . . . . . . 19 (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
795793, 794sylibr 237 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
796795sumeq1d 15056 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
797796ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
798746, 797oveq12d 7164 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
799715, 722, 7983eqtrd 2863 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
800694, 799oveq12d 7164 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
801 fzfid 13343 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) ∈ Fin)
802185adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
803790, 752syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
804 1zzd 12008 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
805803, 804zsubcld 12087 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
806802, 805bccld 41813 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
807806nn0cnd 11952 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
808807adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
809808adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
810 simpl 486 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋))
811 fzelp1 12961 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
812811adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
813810, 812, 567syl2anc 587 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
814812, 605syldan 594 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
815813, 814mulcld 10655 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
816809, 815mulcld 10655 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
817801, 816fsumcl 15088 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
818185adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
819 elfzelz 12909 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
820819adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
821818, 820bccld 41813 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
822821nn0cnd 11952 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
823822adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
824823adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
825 simpll 766 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑𝑖 ∈ (0..^𝑁)))
826 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥𝑋)
827785ssriv 3957 . . . . . . . . . . . . . . . . . . . 20 (1...𝑖) ⊆ (0...𝑖)
828 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
829827, 828sseldi 3951 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
830829adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
831825, 826, 830, 445syl21anc 836 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
832830, 447syldan 594 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
833831, 832mulcld 10655 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
834824, 833mulcld 10655 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
835801, 834fsumcl 15088 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
836678, 817, 744, 835add4d 10862 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘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)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
837 oveq1 7153 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
838837oveq2d 7162 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
839 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐶𝑗) = (𝐶𝑘))
840839fveq1d 6661 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐶𝑗)‘𝑥) = ((𝐶𝑘)‘𝑥))
841 oveq2 7154 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
842841fveq2d 6663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
843842fveq1d 6661 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
844840, 843oveq12d 7164 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
845838, 844oveq12d 7164 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
846 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑘(1...𝑖)
847 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑗(1...𝑖)
848 nfcv 2982 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑖C(𝑗 − 1))
849359, 470nffv 6669 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐶𝑗)‘𝑥)
850588, 470nffv 6669 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
851849, 467, 850nfov 7176 . . . . . . . . . . . . . . . . . . . 20 𝑘(((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
852848, 467, 851nfov 7176 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
853 nfcv 2982 . . . . . . . . . . . . . . . . . . 19 𝑗((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
854845, 846, 847, 852, 853cbvsum 15050 . . . . . . . . . . . . . . . . . 18 Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
855854a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
856855oveq1d 7161 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
857 peano2zm 12020 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
858820, 857syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
859818, 858bccld 41813 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
860859nn0cnd 11952 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
861860adantll 713 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
862861adantlr 714 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
863862, 833mulcld 10655 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
864801, 863, 834fsumadd 15094 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
865864eqcomd 2830 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
866860, 822addcomd 10836 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
867 bcpasc 13684 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ ℕ0𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
868818, 820, 867syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
869866, 868eqtr2d 2860 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
870869oveq1d 7161 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
871870adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
872871adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
873862, 824, 833adddird 10660 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
874872, 873eqtr2d 2860 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
875874sumeq2dv 15058 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
876856, 865, 8753eqtrd 2863 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
877876oveq2d 7162 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘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) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
878 peano2nn0 11932 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
879818, 878syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
880879, 820bccld 41813 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
881880nn0cnd 11952 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
882881adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
883882adantlr 714 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
884883, 833mulcld 10655 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
885801, 884fsumcl 15088 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
886678, 744, 885addassd 10657 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘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) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
887185, 878syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
888 bcn0 13673 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
889887, 888syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
890889, 719oveq12d 7164 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
891890ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
892891, 745eqtr2d 2860 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
893795ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
894893eqcomd 2830 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
895894sumeq1d 15056 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
896892, 895oveq12d 7164 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
897 nfcv 2982 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1)C0)
898897, 467, 701nfov 7176 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
899197, 878syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
900899, 199bccld 41813 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
901900nn0cnd 11952 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
902901adantll 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
903902adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
904903, 448mulcld 10655 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
905 oveq2 7154 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
906905, 713oveq12d 7164 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
907484, 898, 451, 904, 707, 906fsumsplit1 42080 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
908907eqcomd 2830 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
909896, 908eqtrd 2859 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
910909oveq2d 7162 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
911 bcnn 13675 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
912887, 911syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
913912ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
914913oveq1d 7161 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((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))))
915641adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
916915feq1d 6488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
917676, 916mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
918917adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
919 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑥𝑋)
920918, 919ffvelrnd 6841 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
921662, 920mulcld 10655 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
922921mulid2d 10653 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
923643ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
924914, 922, 9233eqtrrd 2864 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
925 fzdifsuc 12969 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
926704, 925syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
927926sumeq1d 15056 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
928927ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
929924, 928oveq12d 7164 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘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) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
930 nfcv 2982 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖 + 1)C(𝑖 + 1))
931651, 470nffv 6669 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐶‘(𝑖 + 1))‘𝑥)
932 nfcv 2982 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑖 + 1) − (𝑖 + 1))
933473, 932nffv 6669 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
934933, 470nffv 6669 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
935931, 467, 934nfov 7176 . . . . . . . . . . . . . . . . . 18 𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
936930, 467, 935nfov 7176 . . . . . . . . . . . . . . . . 17 𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
937 fzfid 13343 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...(𝑖 + 1)) ∈ Fin)
938887adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
939 elfzelz 12909 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
940939adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
941938, 940bccld 41813 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
942941nn0cnd 11952 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
943942adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
944943adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
945646adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
94690a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
947206adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
948946, 947, 9403jca 1125 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
949 elfzle1 12912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
950949adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
951940zred 12082 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
952938nn0red 11951 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
953212adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
954 elfzle2 12913 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
955954adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
956308adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
957951, 952, 953, 955, 956letrd 10791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘𝑁)
958948, 950, 957jca32 519 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
959958, 223sylibr 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
960959adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
961945, 960, 231syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
962961adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
963 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥𝑋)
964962, 963ffvelrnd 6841 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
965945adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
966609adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
967966, 940zsubcld 12087 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
968946, 947, 9673jca 1125 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
969952, 951subge0d 11224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
970955, 969mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘))
971952, 951resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
972953, 951resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁𝑘) ∈ ℝ)
973953, 171, 250sylancl 589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ)
974952, 953, 951, 956lesub1dd 11250 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
975171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℝ)
976975, 951, 953, 950lesub2dd 11251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁𝑘) ≤ (𝑁 − 0))
977971, 972, 973, 974, 976letrd 10791 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0))
978256adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁)
979977, 978breqtrd 5079 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
980968, 970, 979jca32 519 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
981980, 314sylibr 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
982981adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
983982adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
984 fveq2 6659 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷𝑗) = (𝐷‘((𝑖 + 1) − 𝑘)))
985984feq1d 6488 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
986319, 985imbi12d 348 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) −