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Theorem dvnmul 40476
 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 11760 . . . . 5 0 = (ℤ‘0)
42, 3syl6eleq 2740 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12387 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
7 eleq1 2718 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8 fveq2 6229 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9 oveq2 6698 . . . . . . . . . 10 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
109sumeq1d 14475 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
11 oveq1 6697 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 6697 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312fveq2d 6233 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝐷‘(𝑛𝑘)) = (𝐷‘(𝑁𝑘)))
1413fveq1d 6231 . . . . . . . . . . . 12 (𝑛 = 𝑁 → ((𝐷‘(𝑛𝑘))‘𝑥) = ((𝐷‘(𝑁𝑘))‘𝑥))
1514oveq2d 6706 . . . . . . . . . . 11 (𝑛 = 𝑁 → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))
1611, 15oveq12d 6708 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1716sumeq2ad 14478 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1810, 17eqtrd 2685 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1918mpteq2dv 4778 . . . . . . 7 (𝑛 = 𝑁 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))
208, 19eqeq12d 2666 . . . . . 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))))))
2120imbi2d 329 . . . . 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)))))))
227, 21imbi12d 333 . . . 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))))))))
23 fveq2 6229 . . . . . . 7 (𝑚 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0))
24 simpl 472 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑥𝑋) → 𝑚 = 0)
2524oveq2d 6706 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝑥𝑋) → (0...𝑚) = (0...0))
26 simpll 805 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
2726oveq1d 6705 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
2826oveq1d 6705 . . . . . . . . . . . . 13 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚𝑘) = (0 − 𝑘))
2928fveq2d 6233 . . . . . . . . . . . 12 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(0 − 𝑘)))
3029fveq1d 6231 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
3130oveq2d 6706 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
3227, 31oveq12d 6708 . . . . . . . . 9 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3325, 32sumeq12rdv 14482 . . . . . . . 8 ((𝑚 = 0 ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3433mpteq2dva 4777 . . . . . . 7 (𝑚 = 0 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
3523, 34eqeq12d 2666 . . . . . 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))))))
3635imbi2d 329 . . . . 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)))))))
37 fveq2 6229 . . . . . . 7 (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
38 simpl 472 . . . . . . . . . 10 ((𝑚 = 𝑖𝑥𝑋) → 𝑚 = 𝑖)
3938oveq2d 6706 . . . . . . . . 9 ((𝑚 = 𝑖𝑥𝑋) → (0...𝑚) = (0...𝑖))
40 simpll 805 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
4140oveq1d 6705 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
4240oveq1d 6705 . . . . . . . . . . . . 13 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚𝑘) = (𝑖𝑘))
4342fveq2d 6233 . . . . . . . . . . . 12 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑖𝑘)))
4443fveq1d 6231 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑖𝑘))‘𝑥))
4544oveq2d 6706 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
4641, 45oveq12d 6708 . . . . . . . . 9 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4739, 46sumeq12rdv 14482 . . . . . . . 8 ((𝑚 = 𝑖𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4847mpteq2dva 4777 . . . . . . 7 (𝑚 = 𝑖 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))))
4937, 48eqeq12d 2666 . . . . . 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))))))
5049imbi2d 329 . . . . 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)))))))
51 fveq2 6229 . . . . . . 7 (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
52 simpl 472 . . . . . . . . . 10 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → 𝑚 = (𝑖 + 1))
5352oveq2d 6706 . . . . . . . . 9 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
54 simpll 805 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
5554oveq1d 6705 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
5654oveq1d 6705 . . . . . . . . . . . . 13 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚𝑘) = ((𝑖 + 1) − 𝑘))
5756fveq2d 6233 . . . . . . . . . . . 12 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
5857fveq1d 6231 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
5958oveq2d 6706 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
6055, 59oveq12d 6708 . . . . . . . . 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))))
6153, 60sumeq12rdv 14482 . . . . . . . 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))))
6261mpteq2dva 4777 . . . . . . 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)))))
6351, 62eqeq12d 2666 . . . . . 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))))))
6463imbi2d 329 . . . . 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)))))))
65 fveq2 6229 . . . . . . 7 (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
66 simpl 472 . . . . . . . . . 10 ((𝑚 = 𝑛𝑥𝑋) → 𝑚 = 𝑛)
6766oveq2d 6706 . . . . . . . . 9 ((𝑚 = 𝑛𝑥𝑋) → (0...𝑚) = (0...𝑛))
68 simpll 805 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
6968oveq1d 6705 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
7068oveq1d 6705 . . . . . . . . . . . . 13 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚𝑘) = (𝑛𝑘))
7170fveq2d 6233 . . . . . . . . . . . 12 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑛𝑘)))
7271fveq1d 6231 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑛𝑘))‘𝑥))
7372oveq2d 6706 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))
7469, 73oveq12d 6708 . . . . . . . . 9 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7567, 74sumeq12rdv 14482 . . . . . . . 8 ((𝑚 = 𝑛𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7675mpteq2dva 4777 . . . . . . 7 (𝑚 = 𝑛 → (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)))) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))))
7765, 76eqeq12d 2666 . . . . . 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))))))
7877imbi2d 329 . . . . 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)))))))
79 dvnmul.s . . . . . . . . 9 (𝜑𝑆 ∈ {ℝ, ℂ})
80 recnprss 23713 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
8179, 80syl 17 . . . . . . . 8 (𝜑𝑆 ⊆ ℂ)
82 dvnmul.a . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
83 dvnmul.cc . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)
8482, 83mulcld 10098 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) ∈ ℂ)
85 restsspw 16139 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
86 dvnmul.x . . . . . . . . . . 11 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
8785, 86sseldi 3634 . . . . . . . . . 10 (𝜑𝑋 ∈ 𝒫 𝑆)
88 elpwi 4201 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑆𝑋𝑆)
8987, 88syl 17 . . . . . . . . 9 (𝜑𝑋𝑆)
90 cnex 10055 . . . . . . . . . 10 ℂ ∈ V
9190a1i 11 . . . . . . . . 9 (𝜑 → ℂ ∈ V)
9284, 89, 91, 79mptelpm 39671 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
93 dvn0 23732 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
9481, 92, 93syl2anc 694 . . . . . . 7 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
95 0z 11426 . . . . . . . . . . . 12 0 ∈ ℤ
96 fzsn 12421 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
9795, 96ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
9897sumeq1i 14472 . . . . . . . . . 10 Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
9998a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
100 nfcvd 2794 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑘(𝐴 · 𝐵))
101 nfv 1883 . . . . . . . . . 10 𝑘(𝜑𝑥𝑋)
102 oveq2 6698 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C𝑘) = (0C0))
103 0nn0 11345 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
104 bcn0 13137 . . . . . . . . . . . . . . . 16 (0 ∈ ℕ0 → (0C0) = 1)
105103, 104ax-mp 5 . . . . . . . . . . . . . . 15 (0C0) = 1
106105a1i 11 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C0) = 1)
107102, 106eqtrd 2685 . . . . . . . . . . . . 13 (𝑘 = 0 → (0C𝑘) = 1)
108107adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
109 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐶𝑘) = (𝐶‘0))
110109adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶𝑘) = (𝐶‘0))
111 dvnmul.c . . . . . . . . . . . . . . . . . . . . . 22 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
112 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
113112cbvmptv 4783 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
114111, 113eqtri 2673 . . . . . . . . . . . . . . . . . . . . 21 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
115114a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)))
116 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
117116adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
118 eluzfz1 12386 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
1194, 118syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 ∈ (0...𝑁))
120 fvexd 6241 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
121115, 117, 119, 120fvmptd 6327 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
122121adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
123110, 122eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘0))
124 dvnmulf . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑥𝑋𝐴)
12582, 89, 91, 79mptelpm 39671 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
126124, 125syl5eqel 2734 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
127 dvn0 23732 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
12881, 126, 127syl2anc 694 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
129128adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
130123, 129eqtrd 2685 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 0) → (𝐶𝑘) = 𝐹)
131130fveq1d 6231 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
132131adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
133 simpr 476 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
134124fvmpt2 6330 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝐴 ∈ ℂ) → (𝐹𝑥) = 𝐴)
135133, 82, 134syl2anc 694 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐹𝑥) = 𝐴)
136135adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐹𝑥) = 𝐴)
137132, 136eqtrd 2685 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = 𝐴)
138 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
139 0m0e0 11168 . . . . . . . . . . . . . . . . . . . 20 (0 − 0) = 0
140139a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 0) = 0)
141138, 140eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (0 − 𝑘) = 0)
142141fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
143142fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
144143adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
145144adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
146 dvnmul.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
147 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
148147cbvmptv 4783 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
149146, 148eqtri 2673 . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
150149fveq1i 6230 . . . . . . . . . . . . . . . . . 18 (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)
151150a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0))
152 eqidd 2652 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
153 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
154153adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
155 dvnmul.f . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑋𝐵)
15683, 89, 91, 79mptelpm 39671 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝑋𝐵) ∈ (ℂ ↑pm 𝑆))
157155, 156syl5eqel 2734 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐺 ∈ (ℂ ↑pm 𝑆))
158 dvn0 23732 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
15981, 157, 158syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
160159adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
161154, 160eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
162155a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 = (𝑥𝑋𝐵))
163 mptexg 6525 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ 𝒫 𝑆 → (𝑥𝑋𝐵) ∈ V)
16487, 163syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝑋𝐵) ∈ V)
165162, 164eqeltrd 2730 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ V)
166152, 161, 119, 165fvmptd 6327 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
167151, 166eqtrd 2685 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐷‘0) = 𝐺)
168167fveq1d 6231 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
169168ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
170162, 83fvmpt2d 6332 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐺𝑥) = 𝐵)
171170adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐺𝑥) = 𝐵)
172145, 169, 1713eqtrd 2689 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
173137, 172oveq12d 6708 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
174108, 173oveq12d 6708 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
17584mulid2d 10096 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
176175adantr 480 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
177174, 176eqtrd 2685 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
178 0re 10078 . . . . . . . . . . 11 0 ∈ ℝ
179178a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 ∈ ℝ)
180100, 101, 177, 179, 84sumsnd 39499 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
18199, 180eqtr2d 2686 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
182181mpteq2dva 4777 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
18394, 182eqtrd 2685 . . . . . 6 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
184183a1i 11 . . . . 5 (𝑁 ∈ (ℤ‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
185 simp3 1083 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑)
186 simp1 1081 . . . . . . 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))
187 simp2 1082 . . . . . . . 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))))))
188 pm3.35 610 . . . . . . . 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)))))
189185, 187, 188syl2anc 694 . . . . . . 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)))))
19081adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
19192adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
192 elfzonn0 12552 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
193192adantl 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
194 dvnp1 23733 . . . . . . . . . 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)))
195190, 191, 193, 194syl3anc 1366 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
196195adantr 480 . . . . . . . 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)))
197 simpr 476 . . . . . . . . 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)))))
198197oveq2d 6706 . . . . . . . 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))))))
199 eqid 2651 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
200 eqid 2651 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
20179adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
20286adantr 480 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
203 fzfid 12812 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
204192adantr 480 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
205 elfzelz 12380 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
206205adantl 481 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
207204, 206bccld 39844 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
208207nn0cnd 11391 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
209208adantll 750 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
2102093adant3 1101 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
211 simpll 805 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
212 0zd 11427 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
213 elfzoel2 12508 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
214213adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
215212, 214, 2063jca 1261 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
216 elfzle1 12382 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
217216adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
218206zred 11520 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
219213zred 11520 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
220219adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
221192nn0red 11390 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
222221adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
223 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑘𝑖)
224223adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑖)
225 elfzolt2 12518 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
226225adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
227218, 222, 220, 224, 226lelttrd 10233 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
228218, 220, 227ltled 10223 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑁)
229215, 217, 228jca32 557 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
230 elfz2 12371 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
231229, 230sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
232231adantll 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
233 dvnmul.dvnf . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
234111a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
235 fvexd 6241 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
236234, 235fvmpt2d 6332 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
237236feq1d 6068 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐶𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
238233, 237mpbird 247 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ)
239211, 232, 238syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘):𝑋⟶ℂ)
2402393adant3 1101 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐶𝑘):𝑋⟶ℂ)
241 simp3 1083 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → 𝑥𝑋)
242240, 241ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
243192nn0zd 11518 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
244243adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
245244, 206zsubcld 11525 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℤ)
246212, 214, 2453jca 1261 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ))
247 elfzel2 12378 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
248247zred 11520 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
249205zred 11520 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
250248, 249subge0d 10655 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖𝑘) ↔ 𝑘𝑖))
251223, 250mpbird 247 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖𝑘))
252251adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖𝑘))
253222, 218resubcld 10496 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℝ)
254220, 218resubcld 10496 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ∈ ℝ)
255178a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
256220, 255jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
257 resubcl 10383 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
258256, 257syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
259222, 220, 218, 226ltsub1dd 10677 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁𝑘))
260255, 218, 220, 217lesub2dd 10682 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ (𝑁 − 0))
261253, 254, 258, 259, 260ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁 − 0))
262219recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
263262subid1d 10419 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
264263adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
265261, 264breqtrd 4711 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < 𝑁)
266253, 220, 265ltled 10223 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ≤ 𝑁)
267246, 252, 266jca32 557 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
268 elfz2 12371 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
269267, 268sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
270269adantll 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
271 ovex 6718 . . . . . . . . . . . . . . . . . 18 (𝑖𝑘) ∈ V
272 eleq1 2718 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖𝑘) ∈ (0...𝑁)))
273272anbi2d 740 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁))))
274 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
275274feq1d 6068 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
276273, 275imbi12d 333 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)))
277 nfv 1883 . . . . . . . . . . . . . . . . . . 19 𝑘((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
278 eleq1 2718 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
279278anbi2d 740 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑𝑗 ∈ (0...𝑁))))
280 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
281280feq1d 6068 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
282279, 281imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
283 dvnmul.dvng . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
284277, 282, 283chvar 2298 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
285271, 276, 284vtocl 3290 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
286211, 270, 285syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
287149a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
288 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
289288adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑖𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
290 fvexd 6241 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) ∈ V)
291287, 289, 269, 290fvmptd 6327 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
292291adantll 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
293292feq1d 6068 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
294286, 293mpbird 247 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
2952943adant3 1101 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
296295, 241ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
297242, 296mulcld 10098 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
298210, 297mulcld 10098 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
2992103expa 1284 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
300244peano2zd 11523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
301300, 206zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
302212, 214, 3013jca 1261 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
303 peano2re 10247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
304248, 303syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
305 peano2re 10247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
306249, 305syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
307249ltp1d 10992 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
308 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
309249, 248, 308, 223leadd1dd 10679 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
310249, 306, 304, 307, 309ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
311249, 304, 310ltled 10223 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
312311adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
313222, 303syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
314313, 218subge0d 10655 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
315312, 314mpbird 247 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
316313, 218resubcld 10496 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
317 elfzop1le2 39816 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
318317adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
319313, 220, 218, 318lesub1dd 10681 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
320260, 264breqtrd 4711 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ 𝑁)
321316, 254, 220, 319, 320letrd 10232 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
322302, 315, 321jca32 557 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
323 elfz2 12371 . . . . . . . . . . . . . . . . . . . . 21 (((𝑖 + 1) − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
324322, 323sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
325324adantll 750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
326 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) − 𝑘) ∈ V
327 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
328327anbi2d 740 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
329 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
330329feq1d 6068 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
331328, 330imbi12d 333 . . . . . . . . . . . . . . . . . . . 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)))
332326, 331, 284vtocl 3290 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
333211, 325, 332syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
334149a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
335 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
336335fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
337 fvexd 6241 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
338334, 336, 325, 337fvmptd 6327 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
339338feq1d 6068 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
340333, 339mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
341340ffvelrnda 6399 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
3422423expa 1284 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
343341, 342mulcomd 10099 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
344343oveq2d 6706 . . . . . . . . . . . . . 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))))
345206peano2zd 11523 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
346212, 214, 3453jca 1261 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ))
347178a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
348347, 249, 306, 216, 307lelttrd 10233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
349347, 306, 348ltled 10223 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
350349adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
351218, 305syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
352309adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
353351, 313, 220, 352, 318letrd 10232 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
354346, 350, 353jca32 557 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
355 elfz2 12371 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 + 1) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
356354, 355sylibr 224 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
357356adantll 750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
358 ovex 6718 . . . . . . . . . . . . . . . . . . 19 (𝑘 + 1) ∈ V
359 eleq1 2718 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
360359anbi2d 740 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
361 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝐶𝑗) = (𝐶‘(𝑘 + 1)))
362361feq1d 6068 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝐶𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
363360, 362imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 1) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
364 nfv 1883 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝜑𝑗 ∈ (0...𝑁))
365 nfmpt1 4780 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
366111, 365nfcxfr 2791 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝐶
367 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑗
368366, 367nffv 6236 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝐶𝑗)
369 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑋
370 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . 22 𝑘
371368, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶𝑗):𝑋⟶ℂ
372364, 371nfim 1865 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
373 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑗 → (𝐶𝑘) = (𝐶𝑗))
374373feq1d 6068 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶𝑗):𝑋⟶ℂ))
375279, 374imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)))
376372, 375, 238chvar 2298 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
377358, 363, 376vtocl 3290 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
378211, 357, 377syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
379378ffvelrnda 6399 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
3802963expa 1284 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
381379, 380mulcld 10098 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
382341, 342mulcld 10098 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) ∈ ℂ)
383381, 382addcld 10097 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) ∈ ℂ)
384344, 383eqeltrrd 2731 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
385299, 384mulcld 10098 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
3863853impa 1278 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
387211, 79syl 17 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
388178a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → 0 ∈ ℝ)
389211, 86syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
390387, 389, 209dvmptconst 40447 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ (𝑖C𝑘))) = (𝑥𝑋 ↦ 0))
3912973expa 1284 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
392211, 232, 236syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
393392eqcomd 2657 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶𝑘))
394239feqmptd 6288 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)))
395393, 394eqtr2d 2686 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
396395oveq2d 6706 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
397387, 80syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
398211, 126syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
399 elfznn0 12471 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
400399adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
401 dvnp1 23733 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
402397, 398, 400, 401syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
403402eqcomd 2657 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
404114a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)))
405 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
406405adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑘 + 1)) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
407 fvexd 6241 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
408404, 406, 357, 407fvmptd 6327 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
409408eqcomd 2657 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
410378feqmptd 6288 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
411409, 410eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
412396, 403, 4113eqtrd 2689 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
413292eqcomd 2657 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) = (𝐷‘(𝑖𝑘)))
414294feqmptd 6288 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)))
415413, 414eqtr2d 2686 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
416415oveq2d 6706 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
417211, 157syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
418 fznn0sub 12411 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → (𝑖𝑘) ∈ ℕ0)
419418adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℕ0)
420 dvnp1 23733 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
421397, 417, 419, 420syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
422421eqcomd 2657 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)))
423222recnd 10106 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
424 1cnd 10094 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
425218recnd 10106 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
426423, 424, 425addsubd 10451 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖𝑘) + 1))
427426eqcomd 2657 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖𝑘) + 1) = ((𝑖 + 1) − 𝑘))
428427fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
429428adantll 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
430338eqcomd 2657 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
431340feqmptd 6288 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
432429, 430, 4313eqtrd 2689 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
433416, 422, 4323eqtrd 2689 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
434387, 342, 379, 412, 380, 341, 433dvmptmul 23769 . . . . . . . . . . . . 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)))))
435387, 299, 388, 390, 391, 383, 434dvmptmul 23769 . . . . . . . . . . . 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)))))
436391mul02d 10272 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = 0)
437344oveq1d 6705 . . . . . . . . . . . . . . . 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)))
438384, 299mulcomd 10099 . . . . . . . . . . . . . . . 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)))))
439437, 438eqtrd 2685 . . . . . . . . . . . . . . 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)))))
440436, 439oveq12d 6708 . . . . . . . . . . . . . 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))))))
441385addid2d 10275 . . . . . . . . . . . . . 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)))))
442440, 441eqtrd 2685 . . . . . . . . . . . . 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)))))
443442mpteq2dva 4777 . . . . . . . . . . . 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))))))
444435, 443eqtrd 2685 . . . . . . . . . . 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))))))
445199, 200, 201, 202, 203, 298, 386, 444dvmptfsum 23783 . . . . . . . . . 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))))))
446209adantlr 751 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
447381an32s 863 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
448 anass 682 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)))
449 ancom 465 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) ↔ (𝑥𝑋𝑘 ∈ (0...𝑖)))
450449anbi2i 730 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
451 anass 682 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
452451bicomi 214 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
453450, 452bitri 264 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
454448, 453bitri 264 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
455454imbi1i 338 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ))
456342, 455mpbi 220 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
457454imbi1i 338 . . . . . . . . . . . . . . . 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))
458341, 457mpbi 220 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
459456, 458mulcld 10098 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
460446, 447, 459adddid 10102 . . . . . . . . . . . . 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)))))
461460sumeq2dv 14477 . . . . . . . . . . . 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)))))
462203adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...𝑖) ∈ Fin)
463446, 447mulcld 10098 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
464446, 459mulcld 10098 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
465462, 463, 464fsumadd 14514 . . . . . . . . . . . 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)))))
466 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (𝑖C𝑘) = (𝑖C))
467 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 + 1) = ( + 1))
468467fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐶‘(𝑘 + 1)) = (𝐶‘( + 1)))
469468fveq1d 6231 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘( + 1))‘𝑥))
470 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑖𝑘) = (𝑖))
471470fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐷‘(𝑖𝑘)) = (𝐷‘(𝑖)))
472471fveq1d 6231 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐷‘(𝑖𝑘))‘𝑥) = ((𝐷‘(𝑖))‘𝑥))
473469, 472oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) = (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
474466, 473oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))))
475 nfcv 2793 . . . . . . . . . . . . . . . . . 18 (0...𝑖)
476 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑘(0...𝑖)
477 nfcv 2793 . . . . . . . . . . . . . . . . . 18 ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
478 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑘(𝑖C)
479 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑘 ·
480 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . 22 𝑘( + 1)
481366, 480nffv 6236 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶‘( + 1))
482 nfcv 2793 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑥
483481, 482nffv 6236 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐶‘( + 1))‘𝑥)
484 nfmpt1 4780 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
485146, 484nfcxfr 2791 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐷
486 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝑖)
487485, 486nffv 6236 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐷‘(𝑖))
488487, 482nffv 6236 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐷‘(𝑖))‘𝑥)
489483, 479, 488nfov 6716 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))
490478, 479, 489nfov 6716 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
491474, 475, 476, 477, 490cbvsum 14469 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
492491a1i 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))))
493 1zzd 11446 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 1 ∈ ℤ)
49495a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ ℤ)
495243ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑖 ∈ ℤ)
496 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
497 nfcv 2793 . . . . . . . . . . . . . . . . . . . . 21 𝑘
498497, 476nfel 2806 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...𝑖)
499496, 498nfan 1868 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))
500490, 370nfel 2806 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ
501499, 500nfim 1865 . . . . . . . . . . . . . . . . . 18 𝑘((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
502 eleq1 2718 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 ∈ (0...𝑖) ↔ ∈ (0...𝑖)))
503502anbi2d 740 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))))
504474eleq1d 2715 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ))
505503, 504imbi12d 333 . . . . . . . . . . . . . . . . . 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)))
506501, 505, 463chvar 2298 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
507 oveq2 6698 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (𝑖C) = (𝑖C(𝑗 − 1)))
508 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → ( + 1) = ((𝑗 − 1) + 1))
509508fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐶‘( + 1)) = (𝐶‘((𝑗 − 1) + 1)))
510509fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐶‘( + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
511 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → (𝑖) = (𝑖 − (𝑗 − 1)))
512511fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐷‘(𝑖)) = (𝐷‘(𝑖 − (𝑗 − 1))))
513512fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐷‘(𝑖))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
514510, 513oveq12d 6708 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
515507, 514oveq12d 6708 . . . . . . . . . . . . . . . . 17 ( = (𝑗 − 1) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
516493, 494, 495, 506, 515fsumshft 14556 . . . . . . . . . . . . . . . 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))))
517492, 516eqtrd 2685 . . . . . . . . . . . . . . 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))))
518 0p1e1 11170 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
519518oveq1i 6700 . . . . . . . . . . . . . . . . 17 ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
520519sumeq1i 14472 . . . . . . . . . . . . . . . 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)))
521520a1i 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))))
522 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
523522zcnd 11521 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
524 1cnd 10094 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
525523, 524npcand 10434 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
526525fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶𝑗))
527526fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
528527adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
529221recnd 10106 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
530529adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
531523adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
532524adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
533530, 531, 532subsub3d 10460 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
534533fveq2d 6233 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
535534fveq1d 6231 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
536528, 535oveq12d 6708 . . . . . . . . . . . . . . . . . . 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)))
537536oveq2d 6706 . . . . . . . . . . . . . . . . . 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))))
538537sumeq2dv 14477 . . . . . . . . . . . . . . . . 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))))
539538ad2antlr 763 . . . . . . . . . . . . . . . 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))))
540 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑗((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
541 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
542 fzfid 12812 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...(𝑖 + 1)) ∈ Fin)
543192adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
544522adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
545 1zzd 11446 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
546544, 545zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
547543, 546bccld 39844 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
548547nn0cnd 11391 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
549548adantll 750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
550549adantlr 751 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
5511ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
552 0zd 11427 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
553213adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
554552, 553, 5443jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ))
555178a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
556522zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
557 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
558 0lt1 10588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 < 1
559558a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
560 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
561555, 557, 556, 559, 560ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
562555, 556, 561ltled 10223 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
563562adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
564556adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
565221adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
566 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
567565, 566readdcld 10107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
568219adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
569 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
570569adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
571317adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
572564, 567, 568, 570, 571letrd 10232 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗𝑁)
573554, 563, 572jca32 557 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
574 elfz2 12371 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
575573, 574sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
576575adantll 750 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
577551, 576, 376syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
578577adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
579 simplr 807 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥𝑋)
580578, 579ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
581243adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
582581peano2zd 11523 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
583582, 544zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
584552, 553, 5833jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ))
585567, 564subge0d 10655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
586570, 585mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
587567, 564resubcld 10496 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
588587leidd 10632 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
589556, 561elrpd 11907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
590589adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
591567, 590ltsubrpd 11942 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
592587, 567, 568, 591, 571ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
593587, 587, 568, 588, 592lelttrd 10233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
594587, 568, 593ltled 10223 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
595584, 586, 594jca32 557 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
596 elfz2 12371 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 + 1) − 𝑗) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
597595, 596sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
598597adantll 750 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
599 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
600 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘((𝑖 + 1) − 𝑗)
601485, 600nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘((𝑖 + 1) − 𝑗))
602601, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
603599, 602nfim 1865 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
604 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 + 1) − 𝑗) ∈ V
605 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
606605anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
607 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
608607feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
609606, 608imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
610146a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
611 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
612610, 611fvmpt2d 6332 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
613612feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐷𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
614283, 613mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ)
615603, 604, 609, 614vtoclf 3289 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
616551, 598, 615syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
617616adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
618617, 579ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
619580, 618mulcld 10098 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
620550, 619mulcld 10098 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
621 1zzd 11446 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
622243peano2zd 11523 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
623518eqcomi 2660 . . . . . . . . . . . . . . . . . . . . . . 23 1 = (0 + 1)
624623a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
625178a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
626 1red 10093 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
627192nn0ge0d 11392 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
628625, 221, 626, 627leadd1dd 10679 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
629624, 628eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
630621, 622, 6293jca 1261 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
631 eluz2 11731 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
632630, 631sylibr 224 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ‘1))
633 eluzfz2 12387 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
634632, 633syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
635634ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
636 oveq1 6697 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
637636oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
638 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐶𝑗) = (𝐶‘(𝑖 + 1)))
639638fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐶𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
640 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
641640fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
642641fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
643639, 642oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
644637, 643oveq12d 6708 . . . . . . . . . . . . . . . . 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))))
645540, 541, 542, 620, 635, 644fsumsplit1 40122 . . . . . . . . . . . . . . . 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)))))
646 1cnd 10094 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
647529, 646pncand 10431 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
648647oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
649 bcnn 13139 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
650192, 649syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
651648, 650eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
652529, 646addcld 10097 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
653652subidd 10418 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
654653fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
655654fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
656655oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
657651, 656oveq12d 6708 . . . . . . . . . . . . . . . . . . 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))))
658657ad2antlr 763 . . . . . . . . . . . . . . . . . 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))))
659 simpl 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝜑)
660 fzofzp1 12605 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
661660adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
662 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
663 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝑖 + 1)
664366, 663nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐶‘(𝑖 + 1))
665664, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
666662, 665nfim 1865 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
667 ovex 6718 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 + 1) ∈ V
668 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
669668anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
670 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝐶𝑘) = (𝐶‘(𝑖 + 1)))
671670feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
672669, 671imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
673666, 667, 672, 238vtoclf 3289 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
674659, 661, 673syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
675674ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
676 nfv 1883 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
677 nfcv 2793 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑘0
678485, 677nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝐷‘0)
679678, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘0):𝑋⟶ℂ
680676, 679nfim 1865 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
681 c0ex 10072 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
682 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
683682anbi2d 740 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
684 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝐷𝑘) = (𝐷‘0))
685684feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
686683, 685imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
687680, 681, 686, 614vtoclf 3289 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
6881, 119, 687syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐷‘0):𝑋⟶ℂ)
689688adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
690689ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
691675, 690mulcld 10098 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
692691mulid2d 10096 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
693658, 692eqtrd 2685 . . . . . . . . . . . . . . . . 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)))
694 1m1e0 11127 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 − 1) = 0
695694fveq2i 6232 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘(1 − 1)) = (ℤ‘0)
6963eqcomi 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘0) = ℕ0
697695, 696eqtr2i 2674 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘(1 − 1))
698697a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ‘(1 − 1)))
699192, 698eleqtrd 2732 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘(1 − 1)))
700 fzdifsuc2 39838 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (ℤ‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
701699, 700syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
702701eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
703702sumeq1d 14475 . . . . . . . . . . . . . . . . . 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))))
704703ad2antlr 763 . . . . . . . . . . . . . . . . 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))))
705693, 704oveq12d 6708 . . . . . . . . . . . . . . . 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)))))
706539, 645, 7053eqtrd 2689 . . . . . . . . . . . . . . 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)))))
707517, 521, 7063eqtrd 2689 . . . . . . . . . . . . . 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)))))
708 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘(𝑖C0)
709366, 677nffv 6236 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐶‘0)
710709, 482nffv 6236 . . . . . . . . . . . . . . . . . 18 𝑘((𝐶‘0)‘𝑥)
711 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1) − 0)
712485, 711nffv 6236 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐷‘((𝑖 + 1) − 0))
713712, 482nffv 6236 . . . . . . . . . . . . . . . . . 18 𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
714710, 479, 713nfov 6716 . . . . . . . . . . . . . . . . 17 𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
715708, 479, 714nfov 6716 . . . . . . . . . . . . . . . 16 𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
716696a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (ℤ‘0) = ℕ0)
717192, 716eleqtrrd 2733 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘0))
718 eluzfz1 12386 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ‘0) → 0 ∈ (0...𝑖))
719717, 718syl 17 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
720719ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ (0...𝑖))
721 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
722109fveq1d 6231 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐶𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
723 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
724723fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
725724fveq1d 6231 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
726722, 725oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
727721, 726oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
728496, 715, 462, 464, 720, 727fsumsplit1 40122 . . . . . . . . . . . . . . 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)))))
729652subid1d 10419 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
730729fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
731730fveq1d 6231 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
732731oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
733732oveq2d 6706 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
734733oveq1d 6705 . . . . . . . . . . . . . . . 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)))))
735734ad2antlr 763 . . . . . . . . . . . . . . 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)))))
736 bcn0 13137 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
737192, 736syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
738737oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
739738ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
740709, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘0):𝑋⟶ℂ
741676, 740nfim 1865 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
742109feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
743683, 742imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
744741, 681, 743, 238vtoclf 3289 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
7451, 119, 744syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐶‘0):𝑋⟶ℂ)
746745adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
747746ffvelrnda 6399 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
748485, 663nffv 6236 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘(𝑖 + 1))
749748, 369, 370nff 6079 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
750662, 749nfim 1865 . . . . . . . . . . . . . . . . . . . . . 22 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
751 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → (𝐷𝑘) = (𝐷‘(𝑖 + 1)))
752751feq1d 6068 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
753669, 752imbi12d 333 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
754750, 667, 753, 614vtoclf 3289 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
755659, 661, 754syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
756755ffvelrnda 6399 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
757747, 756mulcld 10098 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
758757mulid2d 10096 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
759739, 758eqtrd 2685 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
760 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑖 ∈ (0..^𝑁)
761 1zzd 11446 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
762243adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
763 eldifi 3765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
764 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
765763, 764syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
766765adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
767761, 762, 7663jca 1261 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → (1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
768 elfznn0 12471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
769763, 768syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
770 eldifsni 4353 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
771769, 770jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0𝑗 ≠ 0))
772 elnnne0 11344 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0𝑗 ≠ 0))
773771, 772sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
774 nnge1 11084 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ → 1 ≤ 𝑗)
775773, 774syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
776775adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
777 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
778763, 777syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗𝑖)
779778adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗𝑖)
780767, 776, 779jca32 557 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
781 elfz2 12371 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) ↔ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
782780, 781sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
783782ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
784 0zd 11427 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
785 elfzel2 12378 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
786 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
787784, 785, 7863jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → (0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
788178a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
789786zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
790 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
791558a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 0 < 1)
792 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
793788, 790, 789, 791, 792ltletrd 10235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
794788, 789, 793ltled 10223 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
795 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 𝑗𝑖)
796787, 794, 795jca32 557 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
797 elfz2 12371 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) ↔ ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
798796, 797sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
799788, 793gtned 10210 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
800 nelsn 4245 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
801799, 800syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
802798, 801eldifd 3618 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
803802adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
804803ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
805783, 804impbid 202 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
806760, 805alrimi 2120 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
807 dfcleq 2645 . . . . . . . . . . . . . . . . . . 19 (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
808806, 807sylibr 224 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
809808sumeq1d 14475 . . . . . . . . . . . . . . . . 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))))
810809ad2antlr 763 . . . . . . . . . . . . . . . 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))))
811759, 810oveq12d 6708 . . . . . . . . . . . . . . 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)))))
812728, 735, 8113eqtrd 2689 . . . . . . . . . . . . . 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)))))
813707, 812oveq12d 6708 . . . . . . . . . . . . 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))))))
814 fzfid 12812 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) ∈ Fin)
815192adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
816803, 765syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
817 1zzd 11446 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
818816, 817zsubcld 11525 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
819815, 818bccld 39844 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
820819nn0cnd 11391 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
821820adantll 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
822821adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
823 simpl 472 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋))
824 fzelp1 12431 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
825824adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
826823, 825, 580syl2anc 694 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
827825, 618syldan 486 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
828826, 827mulcld 10098 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
829822, 828mulcld 10098 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
830814, 829fsumcl 14508 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
831192adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
832 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
833832adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
834831, 833bccld 39844 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
835834nn0cnd 11391 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
836835adantll 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
837836adantlr 751 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
838 simpll 805 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑𝑖 ∈ (0..^𝑁)))
839 simplr 807 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥𝑋)
840798ssriv 3640 . . . . . . . . . . . . . . . . . . . 20 (1...𝑖) ⊆ (0...𝑖)
841 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
842840, 841sseldi 3634 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
843842adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
844838, 839, 843, 456syl21anc 1365 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
845843, 458syldan 486 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
846844, 845mulcld 10098 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
847837, 846mulcld 10098 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
848814, 847fsumcl 14508 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
849691, 830, 757, 848add4d 10302 . . . . . . . . . . . . 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))))))
850 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
851850oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
852 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐶𝑗) = (𝐶𝑘))
853852fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐶𝑗)‘𝑥) = ((𝐶𝑘)‘𝑥))
854 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
855854fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
856855fveq1d 6231 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
857853, 856oveq12d 6708 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
858851, 857oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
859 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑘(1...𝑖)
860 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑗(1...𝑖)
861 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑖C(𝑗 − 1))
862368, 482nffv 6236 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐶𝑗)‘𝑥)
863601, 482nffv 6236 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
864862, 479, 863nfov 6716 . . . . . . . . . . . . . . . . . . . 20 𝑘(((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
865861, 479, 864nfov 6716 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
866 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑗((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
867858, 859, 860, 865, 866cbvsum 14469 . . . . . . . . . . . . . . . . . 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)))
868867a1i 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))))
869868oveq1d 6705 . . . . . . . . . . . . . . . 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)))))
870 peano2zm 11458 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
871833, 870syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
872831, 871bccld 39844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
873872nn0cnd 11391 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
874873adantll 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
875874adantlr 751 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
876875, 846mulcld 10098 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
877814, 876, 847fsumadd 14514 . . . . . . . . . . . . . . . . 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)))))
878877eqcomd 2657 . . . . . . . . . . . . . . . 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)))))
879873, 835addcomd 10276 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
880 bcpasc 13148 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ ℕ0𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
881831, 833, 880syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
882879, 881eqtr2d 2686 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
883882oveq1d 6705 . . . . . . . . . . . . . . . . . . . 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))))
884883adantll 750 . . . . . . . . . . . . . . . . . . 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))))
885884adantlr 751 . . . . . . . . . . . . . . . . . 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))))
886875, 837, 846adddird 10103 . . . . . . . . . . . . . . . . . 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)))))
887885, 886eqtr2d 2686 . . . . . . . . . . . . . . . . 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))))
888887sumeq2dv 14477 . . . . . . . . . . . . . . . 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))))
889869, 878, 8883eqtrd 2689 . . . . . . . . . . . . . . 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))))
890889oveq2d 6706 . . . . . . . . . . . . . 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)))))
891 peano2nn0 11371 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
892831, 891syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
893892, 833bccld 39844 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
894893nn0cnd 11391 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
895894adantll 750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
896895adantlr 751 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
897896, 846mulcld 10098 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
898814, 897fsumcl 14508 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
899691, 757, 898addassd 10100 . . . . . . . . . . . . . 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))))))
900192, 891syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
901 bcn0 13137 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
902900, 901syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
903902, 732oveq12d 6708 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
904903ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
905904, 758eqtr2d 2686 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
906808ad2antlr 763 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
907906eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
908907sumeq1d 14475 . . . . . . . . . . . . . . . . . 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))))
909905, 908oveq12d 6708 . . . . . . . . . . . . . . . . 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)))))
910 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1)C0)
911910, 479, 714nfov 6716 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
912204, 891syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
913912, 206bccld 39844 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
914913nn0cnd 11391 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
915914adantll 750 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
916915adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
917916, 459mulcld 10098 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
918 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
919918, 726oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
920496, 911, 462, 917, 720, 919fsumsplit1 40122 . . . . . . . . . . . . . . . . . 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)))))
921920eqcomd 2657 . . . . . . . . . . . . . . . . 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))))
922909, 921eqtrd 2685 . . . . . . . . . . . . . . . 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))))
923922oveq2d 6706 . . . . . . . . . . . . . . 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)))))
924 bcnn 13139 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
925900, 924syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
926925ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
927926oveq1d 6705 . . . . . . . . . . . . . . . . 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))))
928654adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
929928feq1d 6068 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
930689, 929mpbird 247 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
931930adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
932 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑥𝑋)
933931, 932ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
934675, 933mulcld 10098 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
935934mulid2d 10096 . . . . . . . . . . . . . . . . 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)))
936656ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
937927, 935, 9363eqtrrd 2690 . . . . . . . . . . . . . . . 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))))
938 fzdifsuc 12438 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
939717, 938syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
940939sumeq1d 14475 . . . . . . . . . . . . . . . . 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))))
941940ad2antlr 763 . . . . . . . . . . . . . . . 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))))
942937, 941oveq12d 6708 . . . . . . . . . . . . . . 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)))))
943 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖 + 1)C(𝑖 + 1))
944664, 482nffv 6236 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐶‘(𝑖 + 1))‘𝑥)
945 nfcv 2793 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑖 + 1) − (𝑖 + 1))
946485, 945nffv 6236 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
947946, 482nffv 6236 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
948944, 479, 947nfov 6716 . . . . . . . . . . . . . . . . . 18 𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
949943, 479, 948nfov 6716 . . . . . . . . . . . . . . . . 17 𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
950 fzfid 12812 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...(𝑖 + 1)) ∈ Fin)
951900adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
952 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
953952adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
954951, 953bccld 39844 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
955954nn0cnd 11391 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
956955adantll 750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
957956adantlr 751 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
958659adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
95995a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
960213adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
961959, 960, 9533jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
962 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
963962adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
964953zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
965951nn0red 11390 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
966219adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
967 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
968967adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
969317adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
970964, 965, 966, 968, 969letrd 10232 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘𝑁)
971961, 963, 970jca32 557 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
972971, 230sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
973972adantll 750 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
974958, 973, 238syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
975974adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
976 simplr 807 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥𝑋)
977975, 976ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
978958adantlr 751 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
979622adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
980979, 953zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
981959, 960, 9803jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
982965, 964subge0d 10655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
983968, 982mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘))
984965, 964resubcld 10496 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
985966, 964resubcld 10496 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁𝑘) ∈ ℝ)
986966, 178, 257sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ)
987965, 966, 964, 969lesub1dd 10681 . . . . . . . . . . . . . . . . . . . . . . . . . . 27