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Theorem dvnmul 44174
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 12805 . . . . 5 0 = (ℤ‘0)
42, 3eleqtrdi 2848 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 13449 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
7 eleq1 2825 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8 fveq2 6842 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9 oveq2 7365 . . . . . . . . . 10 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
109sumeq1d 15586 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
11 oveq1 7364 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12 fvoveq1 7380 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝐷‘(𝑛𝑘)) = (𝐷‘(𝑁𝑘)))
1312fveq1d 6844 . . . . . . . . . . . 12 (𝑛 = 𝑁 → ((𝐷‘(𝑛𝑘))‘𝑥) = ((𝐷‘(𝑁𝑘))‘𝑥))
1413oveq2d 7373 . . . . . . . . . . 11 (𝑛 = 𝑁 → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))
1511, 14oveq12d 7375 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1615sumeq2sdv 15589 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1710, 16eqtrd 2776 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1817mpteq2dv 5207 . . . . . . 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 2752 . . . . . 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 340 . . . . 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 344 . . . 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 6842 . . . . . . 7 (𝑚 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0))
23 simpl 483 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑥𝑋) → 𝑚 = 0)
2423oveq2d 7373 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝑥𝑋) → (0...𝑚) = (0...0))
25 simpll 765 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
2625oveq1d 7372 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
2725fvoveq1d 7379 . . . . . . . . . . . 12 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(0 − 𝑘)))
2827fveq1d 6844 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
2928oveq2d 7373 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
3026, 29oveq12d 7375 . . . . . . . . 9 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3124, 30sumeq12rdv 15592 . . . . . . . 8 ((𝑚 = 0 ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3231mpteq2dva 5205 . . . . . . 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 2752 . . . . . 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 340 . . . . 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 6842 . . . . . . 7 (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
36 simpl 483 . . . . . . . . . 10 ((𝑚 = 𝑖𝑥𝑋) → 𝑚 = 𝑖)
3736oveq2d 7373 . . . . . . . . 9 ((𝑚 = 𝑖𝑥𝑋) → (0...𝑚) = (0...𝑖))
38 simpll 765 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
3938oveq1d 7372 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
4038fvoveq1d 7379 . . . . . . . . . . . 12 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑖𝑘)))
4140fveq1d 6844 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑖𝑘))‘𝑥))
4241oveq2d 7373 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
4339, 42oveq12d 7375 . . . . . . . . 9 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4437, 43sumeq12rdv 15592 . . . . . . . 8 ((𝑚 = 𝑖𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4544mpteq2dva 5205 . . . . . . 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 2752 . . . . . 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 340 . . . . 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 6842 . . . . . . 7 (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
49 simpl 483 . . . . . . . . . 10 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → 𝑚 = (𝑖 + 1))
5049oveq2d 7373 . . . . . . . . 9 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
51 simpll 765 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
5251oveq1d 7372 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
5351fvoveq1d 7379 . . . . . . . . . . . 12 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
5453fveq1d 6844 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
5554oveq2d 7373 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
5652, 55oveq12d 7375 . . . . . . . . 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 15592 . . . . . . . 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 5205 . . . . . . 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 2752 . . . . . 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 340 . . . . 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 6842 . . . . . . 7 (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
62 simpl 483 . . . . . . . . . 10 ((𝑚 = 𝑛𝑥𝑋) → 𝑚 = 𝑛)
6362oveq2d 7373 . . . . . . . . 9 ((𝑚 = 𝑛𝑥𝑋) → (0...𝑚) = (0...𝑛))
64 simpll 765 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
6564oveq1d 7372 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
6664fvoveq1d 7379 . . . . . . . . . . . 12 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑛𝑘)))
6766fveq1d 6844 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑛𝑘))‘𝑥))
6867oveq2d 7373 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))
6965, 68oveq12d 7375 . . . . . . . . 9 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7063, 69sumeq12rdv 15592 . . . . . . . 8 ((𝑚 = 𝑛𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7170mpteq2dva 5205 . . . . . . 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 2752 . . . . . 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 340 . . . . 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 25268 . . . . . . . . 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 11175 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) ∈ ℂ)
80 restsspw 17313 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
81 dvnmul.x . . . . . . . . . . 11 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
8280, 81sselid 3942 . . . . . . . . . 10 (𝜑𝑋 ∈ 𝒫 𝑆)
83 elpwi 4567 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑆𝑋𝑆)
8482, 83syl 17 . . . . . . . . 9 (𝜑𝑋𝑆)
85 cnex 11132 . . . . . . . . . 10 ℂ ∈ V
8685a1i 11 . . . . . . . . 9 (𝜑 → ℂ ∈ V)
8779, 84, 86, 74mptelpm 43383 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
88 dvn0 25288 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
8976, 87, 88syl2anc 584 . . . . . . 7 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
90 0z 12510 . . . . . . . . . . . 12 0 ∈ ℤ
91 fzsn 13483 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
9290, 91ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
9392sumeq1i 15583 . . . . . . . . . 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 2908 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑘(𝐴 · 𝐵))
96 nfv 1917 . . . . . . . . . 10 𝑘(𝜑𝑥𝑋)
97 oveq2 7365 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C𝑘) = (0C0))
98 0nn0 12428 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
99 bcn0 14210 . . . . . . . . . . . . . . . 16 (0 ∈ ℕ0 → (0C0) = 1)
10098, 99ax-mp 5 . . . . . . . . . . . . . . 15 (0C0) = 1
101100a1i 11 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C0) = 1)
10297, 101eqtrd 2776 . . . . . . . . . . . . 13 (𝑘 = 0 → (0C𝑘) = 1)
103102adantl 482 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
104 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐶𝑘) = (𝐶‘0))
105104adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶𝑘) = (𝐶‘0))
106 dvnmul.c . . . . . . . . . . . . . . . . . . . . 21 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
107 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
108107cbvmptv 5218 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
109106, 108eqtri 2764 . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
110 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
111 eluzfz1 13448 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
1124, 111syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 ∈ (0...𝑁))
113 fvexd 6857 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
114109, 110, 112, 113fvmptd3 6971 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
115114adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
116105, 115eqtrd 2776 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘0))
117 dvnmulf . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑥𝑋𝐴)
11877, 84, 86, 74mptelpm 43383 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
119117, 118eqeltrid 2842 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
120 dvn0 25288 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
12176, 119, 120syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
122121adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
123116, 122eqtrd 2776 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 0) → (𝐶𝑘) = 𝐹)
124123fveq1d 6844 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
125124adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
126 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
127117fvmpt2 6959 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝐴 ∈ ℂ) → (𝐹𝑥) = 𝐴)
128126, 77, 127syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐹𝑥) = 𝐴)
129128adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐹𝑥) = 𝐴)
130125, 129eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = 𝐴)
131 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
132 0m0e0 12273 . . . . . . . . . . . . . . . . . . . 20 (0 − 0) = 0
133132a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 0) = 0)
134131, 133eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (0 − 𝑘) = 0)
135134fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
136135fveq1d 6844 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
137136adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
138137adantlr 713 . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
141140cbvmptv 5218 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
142139, 141eqtri 2764 . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
143142fveq1i 6843 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
146 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
147146adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
148 dvnmul.f . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑋𝐵)
14978, 84, 86, 74mptelpm 43383 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝑋𝐵) ∈ (ℂ ↑pm 𝑆))
150148, 149eqeltrid 2842 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐺 ∈ (ℂ ↑pm 𝑆))
151 dvn0 25288 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
15276, 150, 151syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
153152adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
154147, 153eqtrd 2776 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
155148a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 = (𝑥𝑋𝐵))
156 mptexg 7171 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ 𝒫 𝑆 → (𝑥𝑋𝐵) ∈ V)
15782, 156syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝑋𝐵) ∈ V)
158155, 157eqeltrd 2838 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ V)
159145, 154, 112, 158fvmptd 6955 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
160144, 159eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐷‘0) = 𝐺)
161160fveq1d 6844 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
162161ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
163155, 78fvmpt2d 6961 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐺𝑥) = 𝐵)
164163adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐺𝑥) = 𝐵)
165138, 162, 1643eqtrd 2780 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
166130, 165oveq12d 7375 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
167103, 166oveq12d 7375 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
16879mulid2d 11173 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
169168adantr 481 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
170167, 169eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
171 0re 11157 . . . . . . . . . . 11 0 ∈ ℝ
172171a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 ∈ ℝ)
17395, 96, 170, 172, 79sumsnd 43221 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
17494, 173eqtr2d 2777 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
175174mpteq2dva 5205 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
17689, 175eqtrd 2776 . . . . . 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 1138 . . . . . . 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 1136 . . . . . . 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 1137 . . . . . . . 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 pm3.35 801 . . . . . . . 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 584 . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
18487adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
185 elfzonn0 13617 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
186185adantl 482 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
187 dvnp1 25289 . . . . . . . . . 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 1371 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
189188adantr 481 . . . . . . . 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 485 . . . . . . . . 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 7373 . . . . . . . 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 2736 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
193 eqid 2736 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
19474adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19581adantr 481 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
196 fzfid 13878 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
197185adantr 481 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
198 elfzelz 13441 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
199198adantl 482 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
200197, 199bccld 43539 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
201200nn0cnd 12475 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
202201adantll 712 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
2032023adant3 1132 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
204 simpll 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
205 0zd 12511 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
206 elfzoel2 13571 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
207206adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
208 elfzle1 13444 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
209208adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
210199zred 12607 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
211206zred 12607 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
212211adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
213185nn0red 12474 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
214213adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
215 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 𝑘𝑖)
216215adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑖)
217 elfzolt2 13581 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
218217adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
219210, 214, 212, 216, 218lelttrd 11313 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
220210, 212, 219ltled 11303 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑁)
221205, 207, 199, 209, 220elfzd 13432 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
222221adantll 712 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
223 dvnmul.dvnf . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
224106a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
225 fvexd 6857 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
226224, 225fvmpt2d 6961 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
227226feq1d 6653 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐶𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
228223, 227mpbird 256 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ)
229204, 222, 228syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘):𝑋⟶ℂ)
2302293adant3 1132 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐶𝑘):𝑋⟶ℂ)
231 simp3 1138 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → 𝑥𝑋)
232230, 231ffvelcdmd 7036 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
233185nn0zd 12525 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
234233adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
235234, 199zsubcld 12612 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℤ)
236 elfzel2 13439 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
237236zred 12607 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
238198zred 12607 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
239237, 238subge0d 11745 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖𝑘) ↔ 𝑘𝑖))
240215, 239mpbird 256 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖𝑘))
241240adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖𝑘))
242214, 210resubcld 11583 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℝ)
243212, 210resubcld 11583 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ∈ ℝ)
244171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
245212, 244jca 512 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
246 resubcl 11465 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
247245, 246syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
248214, 212, 210, 218ltsub1dd 11767 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁𝑘))
249244, 210, 212, 209lesub2dd 11772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ (𝑁 − 0))
250242, 243, 247, 248, 249ltletrd 11315 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁 − 0))
251211recnd 11183 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
252251subid1d 11501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
253252adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
254250, 253breqtrd 5131 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < 𝑁)
255242, 212, 254ltled 11303 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ≤ 𝑁)
256205, 207, 235, 241, 255elfzd 13432 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
257256adantll 712 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
258 ovex 7390 . . . . . . . . . . . . . . . . . 18 (𝑖𝑘) ∈ V
259 eleq1 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖𝑘) ∈ (0...𝑁)))
260259anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁))))
261 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
262261feq1d 6653 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
263260, 262imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)))
264 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑘((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
265 eleq1 2825 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
266265anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑𝑗 ∈ (0...𝑁))))
267 fveq2 6842 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
268267feq1d 6653 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
269266, 268imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
270 dvnmul.dvng . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
271264, 269, 270chvarfv 2233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
272258, 263, 271vtocl 3518 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
273204, 257, 272syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
274 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
275 fvexd 6857 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) ∈ V)
276142, 274, 256, 275fvmptd3 6971 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
277276adantll 712 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
278277feq1d 6653 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
279273, 278mpbird 256 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
2802793adant3 1132 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
281280, 231ffvelcdmd 7036 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
282232, 281mulcld 11175 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
283203, 282mulcld 11175 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
2842033expa 1118 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
285234peano2zd 12610 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
286285, 199zsubcld 12612 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
287 peano2re 11328 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
288237, 287syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
289 peano2re 11328 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
290238, 289syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
291238ltp1d 12085 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
292 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
293238, 237, 292, 215leadd1dd 11769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
294238, 290, 288, 291, 293ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
295238, 288, 294ltled 11303 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
296295adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
297214, 287syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
298297, 210subge0d 11745 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
299296, 298mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
300297, 210resubcld 11583 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
301 elfzop1le2 13585 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
302301adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
303297, 212, 210, 302lesub1dd 11771 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
304249, 253breqtrd 5131 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ 𝑁)
305300, 243, 212, 303, 304letrd 11312 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
306205, 207, 286, 299, 305elfzd 13432 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
307306adantll 712 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
308 ovex 7390 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) − 𝑘) ∈ V
309 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
310309anbi2d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
311 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
312311feq1d 6653 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
313310, 312imbi12d 344 . . . . . . . . . . . . . . . . . . . 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)))
314308, 313, 271vtocl 3518 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
315204, 307, 314syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
316142a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
317 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
318317fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
319 fvexd 6857 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
320316, 318, 307, 319fvmptd 6955 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
321320feq1d 6653 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
322315, 321mpbird 256 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
323322ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
3242323expa 1118 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
325323, 324mulcomd 11176 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
326325oveq2d 7373 . . . . . . . . . . . . . 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))))
327199peano2zd 12610 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
328171a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
329328, 238, 290, 208, 291lelttrd 11313 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
330328, 290, 329ltled 11303 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
331330adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
332210, 289syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
333293adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
334332, 297, 212, 333, 302letrd 11312 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
335205, 207, 327, 331, 334elfzd 13432 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
336335adantll 712 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
337 ovex 7390 . . . . . . . . . . . . . . . . . . 19 (𝑘 + 1) ∈ V
338 eleq1 2825 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
339338anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
340 fveq2 6842 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝐶𝑗) = (𝐶‘(𝑘 + 1)))
341340feq1d 6653 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝐶𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
342339, 341imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 1) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
343 nfv 1917 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝜑𝑗 ∈ (0...𝑁))
344 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
345106, 344nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝐶
346 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑗
347345, 346nffv 6852 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝐶𝑗)
348 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑋
349 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑘
350347, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶𝑗):𝑋⟶ℂ
351343, 350nfim 1899 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
352 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑗 → (𝐶𝑘) = (𝐶𝑗))
353352feq1d 6653 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶𝑗):𝑋⟶ℂ))
354266, 353imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)))
355351, 354, 228chvarfv 2233 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
356337, 342, 355vtocl 3518 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
357204, 336, 356syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
358357ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
3592813expa 1118 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
360358, 359mulcld 11175 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
361323, 324mulcld 11175 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) ∈ ℂ)
362360, 361addcld 11174 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) ∈ ℂ)
363326, 362eqeltrrd 2839 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
364284, 363mulcld 11175 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
3653643impa 1110 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
366204, 74syl 17 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
367171a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → 0 ∈ ℝ)
368204, 81syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
369366, 368, 202dvmptconst 44146 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ (𝑖C𝑘))) = (𝑥𝑋 ↦ 0))
3702823expa 1118 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
371204, 222, 226syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
372371eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶𝑘))
373229feqmptd 6910 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)))
374372, 373eqtr2d 2777 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
375374oveq2d 7373 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
376366, 75syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
377204, 119syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
378 elfznn0 13534 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
379378adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
380 dvnp1 25289 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
381376, 377, 379, 380syl3anc 1371 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
382381eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
383 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
384 fvexd 6857 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
385109, 383, 336, 384fvmptd3 6971 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
386385eqcomd 2742 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
387357feqmptd 6910 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
388386, 387eqtrd 2776 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
389375, 382, 3883eqtrd 2780 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
390277eqcomd 2742 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) = (𝐷‘(𝑖𝑘)))
391279feqmptd 6910 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)))
392390, 391eqtr2d 2777 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
393392oveq2d 7373 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
394204, 150syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
395 fznn0sub 13473 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → (𝑖𝑘) ∈ ℕ0)
396395adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℕ0)
397 dvnp1 25289 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
398376, 394, 396, 397syl3anc 1371 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
399398eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)))
400214recnd 11183 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
401 1cnd 11150 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
402210recnd 11183 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
403400, 401, 402addsubd 11533 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖𝑘) + 1))
404403eqcomd 2742 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖𝑘) + 1) = ((𝑖 + 1) − 𝑘))
405404fveq2d 6846 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
406405adantll 712 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
407320eqcomd 2742 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
408322feqmptd 6910 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
409406, 407, 4083eqtrd 2780 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
410393, 399, 4093eqtrd 2780 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
411366, 324, 358, 389, 359, 323, 410dvmptmul 25325 . . . . . . . . . . . . 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)))))
412366, 284, 367, 369, 370, 362, 411dvmptmul 25325 . . . . . . . . . . . 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)))))
413370mul02d 11353 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = 0)
414326oveq1d 7372 . . . . . . . . . . . . . . . 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)))
415363, 284mulcomd 11176 . . . . . . . . . . . . . . . 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)))))
416414, 415eqtrd 2776 . . . . . . . . . . . . . . 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)))))
417413, 416oveq12d 7375 . . . . . . . . . . . . . 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))))))
418364addid2d 11356 . . . . . . . . . . . . . 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)))))
419417, 418eqtrd 2776 . . . . . . . . . . . . 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)))))
420419mpteq2dva 5205 . . . . . . . . . . . 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))))))
421412, 420eqtrd 2776 . . . . . . . . . . 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))))))
422192, 193, 194, 195, 196, 283, 365, 421dvmptfsum 25339 . . . . . . . . . 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))))))
423202adantlr 713 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
424360an32s 650 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
425 anass 469 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)))
426 ancom 461 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) ↔ (𝑥𝑋𝑘 ∈ (0...𝑖)))
427426anbi2i 623 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
428 anass 469 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
429428bicomi 223 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
430427, 429bitri 274 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
431425, 430bitri 274 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
432431imbi1i 349 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ))
433324, 432mpbi 229 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
434431imbi1i 349 . . . . . . . . . . . . . . . 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))
435323, 434mpbi 229 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
436433, 435mulcld 11175 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
437423, 424, 436adddid 11179 . . . . . . . . . . . . 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)))))
438437sumeq2dv 15588 . . . . . . . . . . . 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)))))
439196adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...𝑖) ∈ Fin)
440423, 424mulcld 11175 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
441423, 436mulcld 11175 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
442439, 440, 441fsumadd 15625 . . . . . . . . . . . 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)))))
443 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (𝑖C𝑘) = (𝑖C))
444 fvoveq1 7380 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐶‘(𝑘 + 1)) = (𝐶‘( + 1)))
445444fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘( + 1))‘𝑥))
446 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑖𝑘) = (𝑖))
447446fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐷‘(𝑖𝑘)) = (𝐷‘(𝑖)))
448447fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐷‘(𝑖𝑘))‘𝑥) = ((𝐷‘(𝑖))‘𝑥))
449445, 448oveq12d 7375 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) = (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
450443, 449oveq12d 7375 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))))
451 nfcv 2907 . . . . . . . . . . . . . . . . . 18 (0...𝑖)
452 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑘(0...𝑖)
453 nfcv 2907 . . . . . . . . . . . . . . . . . 18 ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
454 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑘(𝑖C)
455 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑘 ·
456 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑘( + 1)
457345, 456nffv 6852 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶‘( + 1))
458 nfcv 2907 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑥
459457, 458nffv 6852 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐶‘( + 1))‘𝑥)
460 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
461139, 460nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐷
462 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝑖)
463461, 462nffv 6852 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐷‘(𝑖))
464463, 458nffv 6852 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐷‘(𝑖))‘𝑥)
465459, 455, 464nfov 7387 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))
466454, 455, 465nfov 7387 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
467450, 451, 452, 453, 466cbvsum 15580 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
468467a1i 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))))
469 1zzd 12534 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 1 ∈ ℤ)
47090a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ ℤ)
471233ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑖 ∈ ℤ)
472 nfv 1917 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
473 nfcv 2907 . . . . . . . . . . . . . . . . . . . . 21 𝑘
474473, 452nfel 2921 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...𝑖)
475472, 474nfan 1902 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))
476466, 349nfel 2921 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ
477475, 476nfim 1899 . . . . . . . . . . . . . . . . . 18 𝑘((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
478 eleq1 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 ∈ (0...𝑖) ↔ ∈ (0...𝑖)))
479478anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))))
480450eleq1d 2822 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ))
481479, 480imbi12d 344 . . . . . . . . . . . . . . . . . 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)))
482477, 481, 440chvarfv 2233 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
483 oveq2 7365 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (𝑖C) = (𝑖C(𝑗 − 1)))
484 fvoveq1 7380 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐶‘( + 1)) = (𝐶‘((𝑗 − 1) + 1)))
485484fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐶‘( + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
486 oveq2 7365 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → (𝑖) = (𝑖 − (𝑗 − 1)))
487486fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐷‘(𝑖)) = (𝐷‘(𝑖 − (𝑗 − 1))))
488487fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐷‘(𝑖))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
489485, 488oveq12d 7375 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
490483, 489oveq12d 7375 . . . . . . . . . . . . . . . . 17 ( = (𝑗 − 1) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
491469, 470, 471, 482, 490fsumshft 15665 . . . . . . . . . . . . . . . 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))))
492468, 491eqtrd 2776 . . . . . . . . . . . . . . 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))))
493 0p1e1 12275 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
494493oveq1i 7367 . . . . . . . . . . . . . . . . 17 ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
495494sumeq1i 15583 . . . . . . . . . . . . . . . 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)))
496495a1i 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))))
497 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
498497zcnd 12608 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
499 1cnd 11150 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
500498, 499npcand 11516 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
501500fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶𝑗))
502501fveq1d 6844 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
503502adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
504213recnd 11183 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
505504adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
506498adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
507499adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
508505, 506, 507subsub3d 11542 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
509508fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
510509fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
511503, 510oveq12d 7375 . . . . . . . . . . . . . . . . . . 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)))
512511oveq2d 7373 . . . . . . . . . . . . . . . . . 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))))
513512sumeq2dv 15588 . . . . . . . . . . . . . . . . 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))))
514513ad2antlr 725 . . . . . . . . . . . . . . . 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))))
515 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑗((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
516 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
517 fzfid 13878 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...(𝑖 + 1)) ∈ Fin)
518185adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
519497adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
520 1zzd 12534 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
521519, 520zsubcld 12612 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
522518, 521bccld 43539 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
523522nn0cnd 12475 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
524523adantll 712 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
525524adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
5261ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
527 0zd 12511 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
528206adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
529171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
530497zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
531 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
532 0lt1 11677 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 < 1
533532a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
534 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
535529, 531, 530, 533, 534ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
536529, 530, 535ltled 11303 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
537536adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
538530adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
539213adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
540 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
541539, 540readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
542211adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
543 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
544543adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
545301adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
546538, 541, 542, 544, 545letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗𝑁)
547527, 528, 519, 537, 546elfzd 13432 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
548547adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
549526, 548, 355syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
550549adantlr 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
551 simplr 767 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥𝑋)
552550, 551ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
553233adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
554553peano2zd 12610 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
555554, 519zsubcld 12612 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
556541, 538subge0d 11745 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
557544, 556mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
558541, 538resubcld 11583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
559558leidd 11721 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
560530, 535elrpd 12954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
561560adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
562541, 561ltsubrpd 12989 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
563558, 541, 542, 562, 545ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
564558, 558, 542, 559, 563lelttrd 11313 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
565558, 542, 564ltled 11303 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
566527, 528, 555, 557, 565elfzd 13432 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
567566adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
568 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
569 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘((𝑖 + 1) − 𝑗)
570461, 569nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘((𝑖 + 1) − 𝑗))
571570, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
572568, 571nfim 1899 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
573 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 + 1) − 𝑗) ∈ V
574 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
575574anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
576 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
577576feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
578575, 577imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
579139a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
580 fvexd 6857 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
581579, 580fvmpt2d 6961 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
582581feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐷𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
583270, 582mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ)
584572, 573, 578, 583vtoclf 3516 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
585526, 567, 584syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
586585adantlr 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
587586, 551ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
588552, 587mulcld 11175 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
589525, 588mulcld 11175 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
590 1zzd 12534 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
591233peano2zd 12610 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
592493eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 1 = (0 + 1)
593592a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
594171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
595 1red 11156 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
596185nn0ge0d 12476 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
597594, 213, 595, 596leadd1dd 11769 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
598593, 597eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
599590, 591, 5983jca 1128 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
600 eluz2 12769 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
601599, 600sylibr 233 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ‘1))
602 eluzfz2 13449 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
603601, 602syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
604603ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
605 oveq1 7364 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
606605oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
607 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐶𝑗) = (𝐶‘(𝑖 + 1)))
608607fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐶𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
609 oveq2 7365 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
610609fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
611610fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
612608, 611oveq12d 7375 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
613606, 612oveq12d 7375 . . . . . . . . . . . . . . . . 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))))
614515, 516, 517, 589, 604, 613fsumsplit1 15630 . . . . . . . . . . . . . . . 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)))))
615 1cnd 11150 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
616504, 615pncand 11513 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
617616oveq2d 7373 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
618 bcnn 14212 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
619185, 618syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
620617, 619eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
621504, 615addcld 11174 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
622621subidd 11500 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
623622fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
624623fveq1d 6844 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
625624oveq2d 7373 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
626620, 625oveq12d 7375 . . . . . . . . . . . . . . . . . . 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))))
627626ad2antlr 725 . . . . . . . . . . . . . . . . . 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))))
628 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝜑)
629 fzofzp1 13669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
630629adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
631 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
632 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝑖 + 1)
633345, 632nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐶‘(𝑖 + 1))
634633, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
635631, 634nfim 1899 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
636 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 + 1) ∈ V
637 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
638637anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
639 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝐶𝑘) = (𝐶‘(𝑖 + 1)))
640639feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
641638, 640imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
642635, 636, 641, 228vtoclf 3516 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
643628, 630, 642syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
644643ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
645 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
646 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑘0
647461, 646nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝐷‘0)
648647, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘0):𝑋⟶ℂ
649645, 648nfim 1899 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
650 c0ex 11149 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
651 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
652651anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
653 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝐷𝑘) = (𝐷‘0))
654653feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
655652, 654imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
656649, 650, 655, 583vtoclf 3516 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
6571, 112, 656syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐷‘0):𝑋⟶ℂ)
658657adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
659658ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
660644, 659mulcld 11175 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
661660mulid2d 11173 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
662627, 661eqtrd 2776 . . . . . . . . . . . . . . . . 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)))
663 1m1e0 12225 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 − 1) = 0
664663fveq2i 6845 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘(1 − 1)) = (ℤ‘0)
6653eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘0) = ℕ0
666664, 665eqtr2i 2765 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘(1 − 1))
667666a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ‘(1 − 1)))
668185, 667eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘(1 − 1)))
669 fzdifsuc2 43534 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (ℤ‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
670668, 669syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
671670eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
672671sumeq1d 15586 . . . . . . . . . . . . . . . . . 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))))
673672ad2antlr 725 . . . . . . . . . . . . . . . . 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))))
674662, 673oveq12d 7375 . . . . . . . . . . . . . . . 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)))))
675514, 614, 6743eqtrd 2780 . . . . . . . . . . . . . . 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)))))
676492, 496, 6753eqtrd 2780 . . . . . . . . . . . . . 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)))))
677 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘(𝑖C0)
678345, 646nffv 6852 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐶‘0)
679678, 458nffv 6852 . . . . . . . . . . . . . . . . . 18 𝑘((𝐶‘0)‘𝑥)
680 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1) − 0)
681461, 680nffv 6852 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐷‘((𝑖 + 1) − 0))
682681, 458nffv 6852 . . . . . . . . . . . . . . . . . 18 𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
683679, 455, 682nfov 7387 . . . . . . . . . . . . . . . . 17 𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
684677, 455, 683nfov 7387 . . . . . . . . . . . . . . . 16 𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
685665a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (ℤ‘0) = ℕ0)
686185, 685eleqtrrd 2841 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘0))
687 eluzfz1 13448 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ‘0) → 0 ∈ (0...𝑖))
688686, 687syl 17 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
689688ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ (0...𝑖))
690 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
691104fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐶𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
692 oveq2 7365 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
693692fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
694693fveq1d 6844 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
695691, 694oveq12d 7375 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
696690, 695oveq12d 7375 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
697472, 684, 439, 441, 689, 696fsumsplit1 15630 . . . . . . . . . . . . . . 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)))))
698621subid1d 11501 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
699698fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
700699fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
701700oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
702701oveq2d 7373 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
703702oveq1d 7372 . . . . . . . . . . . . . . . 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)))))
704703ad2antlr 725 . . . . . . . . . . . . . . 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)))))
705 bcn0 14210 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
706185, 705syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
707706oveq1d 7372 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
708707ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
709678, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘0):𝑋⟶ℂ
710645, 709nfim 1899 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
711104feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
712652, 711imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
713710, 650, 712, 228vtoclf 3516 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
7141, 112, 713syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐶‘0):𝑋⟶ℂ)
715714adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
716715ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
717461, 632nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘(𝑖 + 1))
718717, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
719631, 718nfim 1899 . . . . . . . . . . . . . . . . . . . . . 22 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
720 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → (𝐷𝑘) = (𝐷‘(𝑖 + 1)))
721720feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
722638, 721imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
723719, 636, 722, 583vtoclf 3516 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
724628, 630, 723syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
725724ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
726716, 725mulcld 11175 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
727726mulid2d 11173 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
728708, 727eqtrd 2776 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
729 nfv 1917 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑖 ∈ (0..^𝑁)
730 1zzd 12534 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
731233adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
732 eldifi 4086 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
733 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
734732, 733syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
735734adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
736 elfznn0 13534 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
737732, 736syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
738 eldifsni 4750 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
739737, 738jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0𝑗 ≠ 0))
740 elnnne0 12427 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0𝑗 ≠ 0))
741739, 740sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
742 nnge1 12181 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ℕ → 1 ≤ 𝑗)
743741, 742syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
744743adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
745 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
746732, 745syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗𝑖)
747746adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗𝑖)
748730, 731, 735, 744, 747elfzd 13432 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
749748ex 413 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
750 0zd 12511 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
751 elfzel2 13439 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
752 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
753171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
754752zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
755 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
756532a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 < 1)
757 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
758753, 755, 754, 756, 757ltletrd 11315 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
759753, 754, 758ltled 11303 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
760 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗𝑖)
761750, 751, 752, 759, 760elfzd 13432 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
762753, 758gtned 11290 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
763 nelsn 4626 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
764762, 763syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
765761, 764eldifd 3921 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
766765adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
767766ex 413 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
768749, 767impbid 211 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
769729, 768alrimi 2206 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
770 dfcleq 2729 . . . . . . . . . . . . . . . . . . 19 (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
771769, 770sylibr 233 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
772771sumeq1d 15586 . . . . . . . . . . . . . . . . 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))))
773772ad2antlr 725 . . . . . . . . . . . . . . . 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))))
774728, 773oveq12d 7375 . . . . . . . . . . . . . . 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)))))
775697, 704, 7743eqtrd 2780 . . . . . . . . . . . . . 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)))))
776676, 775oveq12d 7375 . . . . . . . . . . . . 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))))))
777 fzfid 13878 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) ∈ Fin)
778185adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
779766, 734syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
780 1zzd 12534 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
781779, 780zsubcld 12612 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
782778, 781bccld 43539 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
783782nn0cnd 12475 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
784783adantll 712 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
785784adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
786 simpl 483 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋))
787 fzelp1 13493 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
788787adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
789786, 788, 552syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
790788, 587syldan 591 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
791789, 790mulcld 11175 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
792785, 791mulcld 11175 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
793777, 792fsumcl 15618 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
794185adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
795 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
796795adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
797794, 796bccld 43539 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
798797nn0cnd 12475 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
799798adantll 712 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
800799adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
801 simpll 765 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑𝑖 ∈ (0..^𝑁)))
802 simplr 767 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥𝑋)
803761ssriv 3948 . . . . . . . . . . . . . . . . . . . 20 (1...𝑖) ⊆ (0...𝑖)
804 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
805803, 804sselid 3942 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
806805adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
807801, 802, 806, 433syl21anc 836 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
808806, 435syldan 591 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
809807, 808mulcld 11175 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
810800, 809mulcld 11175 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
811777, 810fsumcl 15618 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
812660, 793, 726, 811add4d 11383 . . . . . . . . . . . . 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))))))
813 oveq1 7364 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
814813oveq2d 7373 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
815 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐶𝑗) = (𝐶𝑘))
816815fveq1d 6844 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐶𝑗)‘𝑥) = ((𝐶𝑘)‘𝑥))
817 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
818817fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
819818fveq1d 6844 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
820816, 819oveq12d 7375 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
821814, 820oveq12d 7375 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
822 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑘(1...𝑖)
823 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑗(1...𝑖)
824 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑖C(𝑗 − 1))
825347, 458nffv 6852 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐶𝑗)‘𝑥)
826570, 458nffv 6852 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
827825, 455, 826nfov 7387 . . . . . . . . . . . . . . . . . . . 20 𝑘(((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
828824, 455, 827nfov 7387 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
829 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑗((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
830821, 822, 823, 828, 829cbvsum 15580 . . . . . . . . . . . . . . . . . 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)))
831830a1i 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))))
832831oveq1d 7372 . . . . . . . . . . . . . . . 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)))))
833 peano2zm 12546 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
834796, 833syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
835794, 834bccld 43539 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
836835nn0cnd 12475 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
837836adantll 712 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
838837adantlr 713 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
839838, 809mulcld 11175 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
840777, 839, 810fsumadd 15625 . . . . . . . . . . . . . . . . 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)))))
841840eqcomd 2742 . . . . . . . . . . . . . . . 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)))))
842836, 798addcomd 11357 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
843 bcpasc 14221 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ ℕ0𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
844794, 796, 843syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
845842, 844eqtr2d 2777 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
846845oveq1d 7372 . . . . . . . . . . . . . . . . . . . 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))))
847846adantll 712 . . . . . . . . . . . . . . . . . . 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))))
848847adantlr 713 . . . . . . . . . . . . . . . . . 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))))
849838, 800, 809adddird 11180 . . . . . . . . . . . . . . . . . 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)))))
850848, 849eqtr2d 2777 . . . . . . . . . . . . . . . . 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))))
851850sumeq2dv 15588 . . . . . . . . . . . . . . . 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))))
852832, 841, 8513eqtrd 2780 . . . . . . . . . . . . . . 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))))
853852oveq2d 7373 . . . . . . . . . . . . . 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)))))
854 peano2nn0 12453 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
855794, 854syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
856855, 796bccld 43539 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
857856nn0cnd 12475 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
858857adantll 712 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
859858adantlr 713 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
860859, 809mulcld 11175 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
861777, 860fsumcl 15618 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
862660, 726, 861addassd 11177 . . . . . . . . . . . . . 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))))))
863185, 854syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
864 bcn0 14210 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
865863, 864syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
866865, 701oveq12d 7375 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
867866ad2antlr 725 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
868867, 727eqtr2d 2777 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
869771ad2antlr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
870869eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
871870sumeq1d 15586 . . . . . . . . . . . . . . . . . 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))))
872868, 871oveq12d 7375 . . . . . . . . . . . . . . . . 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)))))
873 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1)C0)
874873, 455, 683nfov 7387 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
875197, 854syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
876875, 199bccld 43539 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
877876nn0cnd 12475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
878877adantll 712 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
879878adantlr 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
880879, 436mulcld 11175 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
881 oveq2 7365 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
882881, 695oveq12d 7375 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
883472, 874, 439, 880, 689, 882fsumsplit1 15630 . . . . . . . . . . . . . . . . . 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)))))
884883eqcomd 2742 . . . . . . . . . . . . . . . . 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))))
885872, 884eqtrd 2776 . . . . . . . . . . . . . . . 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))))
886885oveq2d 7373 . . . . . . . . . . . . . . 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)))))
887 bcnn 14212 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
888863, 887syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
889888ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
890889oveq1d 7372 . . . . . . . . . . . . . . . . 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))))
891623adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
892891feq1d 6653 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
893658, 892mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
894893adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
895 simpr 485 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑥𝑋)
896894, 895ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
897644, 896mulcld 11175 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
898897mulid2d 11173 . . . . . . . . . . . . . . . . 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)))
899625ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
900890, 898, 8993eqtrrd 2781 . . . . . . . . . . . . . . . 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))))
901 fzdifsuc 13501 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
902686, 901syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
903902sumeq1d 15586 . . . . . . . . . . . . . . . . 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))))
904903ad2antlr 725 . . . . . . . . . . . . . . . 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))))
905900, 904oveq12d 7375 . . . . . . . . . . . . . . 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)))))
906 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖 + 1)C(𝑖 + 1))
907633, 458nffv 6852 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐶‘(𝑖 + 1))‘𝑥)
908 nfcv 2907 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑖 + 1) − (𝑖 + 1))
909461, 908nffv 6852 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
910909, 458nffv 6852 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
911907, 455, 910nfov 7387 . . . . . . . . . . . . . . . . . 18 𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
912906, 455, 911nfov 7387 . . . . . . . . . . . . . . . . 17 𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
913 fzfid 13878 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...(𝑖 + 1)) ∈ Fin)
914863adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
915 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
916915adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
917914, 916bccld 43539 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
918917nn0cnd 12475 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
919918adantll 712 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
920919adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
921628adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
92290a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
923206adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
924 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
925924adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
926916zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
927914nn0red 12474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
928211adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
929 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
930929adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
931301adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
932926, 927, 928, 930, 931letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘𝑁)
933922, 923, 916, 925, 932elfzd 13432 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
934933adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
935921, 934, 228syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
936935adantlr 713 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
937 simplr 767 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥𝑋)
938936, 937ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
939921adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
940591adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
941940, 916zsubcld 12612 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
942927, 926subge0d 11745 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
943930, 942mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘))
944927, 926resubcld 11583 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
945928, 926resubcld 11583 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁𝑘) ∈ ℝ)
946928, 171, 246sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ)
947927, 928, 926, 931lesub1dd 11771 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
948171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℝ)
949948, 926, 928, 925lesub2dd 11772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁𝑘) ≤ (𝑁 − 0))
950944, 945, 946, 947, 949letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0))
951252adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁)
952950, 951breqtrd 5131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
953922, 923, 941, 943, 952elfzd 13432 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
954953adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
955954adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
956 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷𝑗) = (𝐷‘((𝑖 + 1) − 𝑘)))
957956feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
958310, 957imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
959461, 346nffv 6852 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷𝑗)
960959, 348, 349nff 6664 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷𝑗):𝑋⟶ℂ
961343, 960nfim 1899 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗):𝑋⟶ℂ)
962 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑗 → (𝐷𝑘) = (𝐷𝑗))
963962feq1d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑗 → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷𝑗):𝑋⟶ℂ))
964266, 963imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗):𝑋⟶ℂ)))
965961, 964, 583chvarfv 2233 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐷𝑗):𝑋⟶ℂ)
966308, 958, 965vtocl 3518 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
967939, 955, 966syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
968967, 937ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
969938, 968mulcld 11175 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
970920, 969mulcld 11175 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
971863, 685eleqtrrd 2841 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ‘0))
972 eluzfz2 13449 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ‘0) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
973971, 972syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
974973ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
975 oveq2 7365 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑖 + 1) → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C(𝑖 + 1)))
976639fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑖 + 1) → ((𝐶𝑘)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
977 oveq2 7365 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑖 + 1) → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − (𝑖 + 1)))
978977fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
979978fveq1d 6844 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
980976, 979oveq12d 7375 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑖 + 1) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
981975, 980oveq12d 7375 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑖 + 1) → (((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))))
982472, 912, 913, 970, 974, 981fsumsplit1 15630 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...(𝑖 + 1))(((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)))))
983982eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((((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)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
984886, 905, 9833eqtrd 2780 . . . . . . . . . . . . . 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))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
985853, 862, 9843eqtrd 2780 . . . . . . . . . . . . 13 (((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))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
986776, 812, 9853eqtrd 2780 . . . . . . . . . . . 12 (((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)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
987438, 442, 9863eqtrd 2780 . . . . . . . . . . 11 (((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 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
988987mpteq2dva 5205 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
989422, 988eqtrd 2776 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
990989adantr 481 . . . . . . . 8 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) → (𝑆 D (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
991189, 191, 9903eqtrd 2780 . . . . . . 7 (((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)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
992178, 179, 182, 991syl21anc 836 . . . . . 6 ((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 + 1)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
9939923exp 1119 . . . . 5 (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 + 1)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
99434, 47, 60, 73, 177, 993fzind2 13690 . . . 4 (𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))))
99521, 994vtoclg 3525 . . 3 (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))))
9962, 6, 995sylc 65 . 2 (𝜑 → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))))
9971, 996mpd 15 1 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  cdif 3907  wss 3910  𝒫 cpw 4560  {csn 4586  {cpr 4588   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  pm cpm 8766  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385  cn 12153  0cn0 12413  cz 12499  cuz 12763  +crp 12915  ...cfz 13424  ..^cfzo 13567  Ccbc 14202  Σcsu 15570  t crest 17302  TopOpenctopn 17303  fldccnfld 20796   D cdv 25227   D𝑛 cdvn 25228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-dvn 25232
This theorem is referenced by:  dvnprodlem2  44178
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