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Theorem dvnmul 38657
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 11557 . . . . 5 0 = (ℤ‘0)
42, 3syl6eleq 2697 . . . 4 (𝜑𝑁 ∈ (ℤ‘0))
5 eluzfz2 12178 . . . 4 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑𝑁 ∈ (0...𝑁))
7 eleq1 2675 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8 fveq2 6088 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9 oveq2 6535 . . . . . . . . . 10 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
109sumeq1d 14228 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
11 oveq1 6534 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 6534 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312fveq2d 6092 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝐷‘(𝑛𝑘)) = (𝐷‘(𝑁𝑘)))
1413fveq1d 6090 . . . . . . . . . . . 12 (𝑛 = 𝑁 → ((𝐷‘(𝑛𝑘))‘𝑥) = ((𝐷‘(𝑁𝑘))‘𝑥))
1514oveq2d 6543 . . . . . . . . . . 11 (𝑛 = 𝑁 → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))
1611, 15oveq12d 6545 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1716sumeq2ad 38456 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1810, 17eqtrd 2643 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥))))
1918mpteq2dv 4667 . . . . . . 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 2624 . . . . . 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 328 . . . . 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 332 . . . 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 6088 . . . . . . 7 (𝑚 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0))
24 simpl 471 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑥𝑋) → 𝑚 = 0)
2524oveq2d 6543 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝑥𝑋) → (0...𝑚) = (0...0))
26 simpll 785 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
2726oveq1d 6542 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
2826oveq1d 6542 . . . . . . . . . . . . 13 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚𝑘) = (0 − 𝑘))
2928fveq2d 6092 . . . . . . . . . . . 12 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(0 − 𝑘)))
3029fveq1d 6090 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
3130oveq2d 6543 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
3227, 31oveq12d 6545 . . . . . . . . 9 (((𝑚 = 0 ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3325, 32sumeq12rdv 14234 . . . . . . . 8 ((𝑚 = 0 ∧ 𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3433mpteq2dva 4666 . . . . . . 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 2624 . . . . . 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 328 . . . . 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 6088 . . . . . . 7 (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
38 simpl 471 . . . . . . . . . 10 ((𝑚 = 𝑖𝑥𝑋) → 𝑚 = 𝑖)
3938oveq2d 6543 . . . . . . . . 9 ((𝑚 = 𝑖𝑥𝑋) → (0...𝑚) = (0...𝑖))
40 simpll 785 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
4140oveq1d 6542 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
4240oveq1d 6542 . . . . . . . . . . . . 13 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚𝑘) = (𝑖𝑘))
4342fveq2d 6092 . . . . . . . . . . . 12 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑖𝑘)))
4443fveq1d 6090 . . . . . . . . . . 11 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑖𝑘))‘𝑥))
4544oveq2d 6543 . . . . . . . . . 10 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
4641, 45oveq12d 6545 . . . . . . . . 9 (((𝑚 = 𝑖𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4739, 46sumeq12rdv 14234 . . . . . . . 8 ((𝑚 = 𝑖𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))
4847mpteq2dva 4666 . . . . . . 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 2624 . . . . . 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 328 . . . . 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 6088 . . . . . . 7 (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
52 simpl 471 . . . . . . . . . 10 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → 𝑚 = (𝑖 + 1))
5352oveq2d 6543 . . . . . . . . 9 ((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
54 simpll 785 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
5554oveq1d 6542 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
5654oveq1d 6542 . . . . . . . . . . . . 13 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚𝑘) = ((𝑖 + 1) − 𝑘))
5756fveq2d 6092 . . . . . . . . . . . 12 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
5857fveq1d 6090 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
5958oveq2d 6543 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
6055, 59oveq12d 6545 . . . . . . . . 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 14234 . . . . . . . 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 4666 . . . . . . 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 2624 . . . . . 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 328 . . . . 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 6088 . . . . . . 7 (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
66 simpl 471 . . . . . . . . . 10 ((𝑚 = 𝑛𝑥𝑋) → 𝑚 = 𝑛)
6766oveq2d 6543 . . . . . . . . 9 ((𝑚 = 𝑛𝑥𝑋) → (0...𝑚) = (0...𝑛))
68 simpll 785 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
6968oveq1d 6542 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
7068oveq1d 6542 . . . . . . . . . . . . 13 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚𝑘) = (𝑛𝑘))
7170fveq2d 6092 . . . . . . . . . . . 12 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚𝑘)) = (𝐷‘(𝑛𝑘)))
7271fveq1d 6090 . . . . . . . . . . 11 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚𝑘))‘𝑥) = ((𝐷‘(𝑛𝑘))‘𝑥))
7372oveq2d 6543 . . . . . . . . . 10 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥)))
7469, 73oveq12d 6545 . . . . . . . . 9 (((𝑚 = 𝑛𝑥𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7567, 74sumeq12rdv 14234 . . . . . . . 8 ((𝑚 = 𝑛𝑥𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑚𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑛𝑘))‘𝑥))))
7675mpteq2dva 4666 . . . . . . 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 2624 . . . . . 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 328 . . . . 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 23419 . . . . . . . . 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 9917 . . . . . . . . 9 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) ∈ ℂ)
85 restsspw 15864 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
86 dvnmul.x . . . . . . . . . . 11 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
8785, 86sseldi 3565 . . . . . . . . . 10 (𝜑𝑋 ∈ 𝒫 𝑆)
88 elpwi 4116 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑆𝑋𝑆)
8987, 88syl 17 . . . . . . . . 9 (𝜑𝑋𝑆)
90 cnex 9874 . . . . . . . . . 10 ℂ ∈ V
9190a1i 11 . . . . . . . . 9 (𝜑 → ℂ ∈ V)
9284, 89, 91, 79mptelpm 38176 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
93 dvn0 23438 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
9481, 92, 93syl2anc 690 . . . . . . 7 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ (𝐴 · 𝐵)))
95 0z 11224 . . . . . . . . . . . 12 0 ∈ ℤ
96 fzsn 12212 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
9795, 96ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
9897sumeq1i 14225 . . . . . . . . . 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 2751 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑘(𝐴 · 𝐵))
101 nfv 1829 . . . . . . . . . 10 𝑘(𝜑𝑥𝑋)
102 oveq2 6535 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C𝑘) = (0C0))
103 0nn0 11157 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
104 bcn0 12917 . . . . . . . . . . . . . . . 16 (0 ∈ ℕ0 → (0C0) = 1)
105103, 104ax-mp 5 . . . . . . . . . . . . . . 15 (0C0) = 1
106105a1i 11 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C0) = 1)
107102, 106eqtrd 2643 . . . . . . . . . . . . 13 (𝑘 = 0 → (0C𝑘) = 1)
108107adantl 480 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
109 fveq2 6088 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐶𝑘) = (𝐶‘0))
110109adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶𝑘) = (𝐶‘0))
111 dvnmul.c . . . . . . . . . . . . . . . . . . . . . 22 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
112 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
113112cbvmptv 4672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
114111, 113eqtri 2631 . . . . . . . . . . . . . . . . . . . . 21 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
115114a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)))
116 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
117116adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
118 eluzfz1 12177 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
1194, 118syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 ∈ (0...𝑁))
120 fvex 6098 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 D𝑛 𝐹)‘0) ∈ V
121120a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
122115, 117, 119, 121fvmptd 6182 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
123122adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
124110, 123eqtrd 2643 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘0))
125 dvnmulf . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑥𝑋𝐴)
12682, 89, 91, 79mptelpm 38176 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
127125, 126syl5eqel 2691 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
128 dvn0 23438 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
12981, 127, 128syl2anc 690 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
130129adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
131124, 130eqtrd 2643 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 0) → (𝐶𝑘) = 𝐹)
132131fveq1d 6090 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
133132adantlr 746 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = (𝐹𝑥))
134 simpr 475 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
135125fvmpt2 6185 . . . . . . . . . . . . . . . 16 ((𝑥𝑋𝐴 ∈ ℂ) → (𝐹𝑥) = 𝐴)
136134, 82, 135syl2anc 690 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐹𝑥) = 𝐴)
137136adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐹𝑥) = 𝐴)
138133, 137eqtrd 2643 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐶𝑘)‘𝑥) = 𝐴)
139 oveq2 6535 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
140 0m0e0 10980 . . . . . . . . . . . . . . . . . . . 20 (0 − 0) = 0
141140a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 0) = 0)
142139, 141eqtrd 2643 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (0 − 𝑘) = 0)
143142fveq2d 6092 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
144143fveq1d 6090 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
145144adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
146145adantlr 746 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
147 dvnmul.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
148 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
149148cbvmptv 4672 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
150147, 149eqtri 2631 . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
151150fveq1i 6089 . . . . . . . . . . . . . . . . . 18 (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)
152151a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0))
153 eqidd 2610 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
154 fveq2 6088 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
155154adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
156 dvnmul.f . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑋𝐵)
15783, 89, 91, 79mptelpm 38176 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥𝑋𝐵) ∈ (ℂ ↑pm 𝑆))
158156, 157syl5eqel 2691 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐺 ∈ (ℂ ↑pm 𝑆))
159 dvn0 23438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
16081, 158, 159syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
161160adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
162155, 161eqtrd 2643 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
163156a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 = (𝑥𝑋𝐵))
164 mptexg 6367 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ 𝒫 𝑆 → (𝑥𝑋𝐵) ∈ V)
16587, 164syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥𝑋𝐵) ∈ V)
166163, 165eqeltrd 2687 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 ∈ V)
167153, 162, 119, 166fvmptd 6182 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
168152, 167eqtrd 2643 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐷‘0) = 𝐺)
169168fveq1d 6090 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
170169ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺𝑥))
171163, 83fvmpt2d 6187 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝐺𝑥) = 𝐵)
172171adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (𝐺𝑥) = 𝐵)
173146, 170, 1723eqtrd 2647 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
174138, 173oveq12d 6545 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
175108, 174oveq12d 6545 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
17684mulid2d 9915 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
177176adantr 479 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
178175, 177eqtrd 2643 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
179 0re 9897 . . . . . . . . . . 11 0 ∈ ℝ
180179a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 ∈ ℝ)
181100, 101, 178, 180, 84sumsnd 38032 . . . . . . . . 9 ((𝜑𝑥𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
18299, 181eqtr2d 2644 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
183182mpteq2dva 4666 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
18494, 183eqtrd 2643 . . . . . 6 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
185184a1i 11 . . . . 5 (𝑁 ∈ (ℤ‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
186 simp3 1055 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑)
187 simp1 1053 . . . . . . 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))
188 simp2 1054 . . . . . . . 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))))))
189 pm3.35 608 . . . . . . . 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)))))
190186, 188, 189syl2anc 690 . . . . . . 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)))))
19181adantr 479 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
19292adantr 479 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
193 elfzonn0 12338 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
194193adantl 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
195 dvnp1 23439 . . . . . . . . . 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)))
196191, 192, 194, 195syl3anc 1317 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
197196adantr 479 . . . . . . . 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)))
198 simpr 475 . . . . . . . . 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)))))
199198oveq2d 6543 . . . . . . . 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))))))
200 eqid 2609 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
201 eqid 2609 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
20279adantr 479 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
20386adantr 479 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
204 fzfid 12592 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
205193adantr 479 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
206 elfzelz 12171 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
207206adantl 480 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
208205, 207bccld 38296 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
209208nn0cnd 11203 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
210209adantll 745 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
2112103adant3 1073 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
212 simpll 785 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
213 0zd 11225 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
214 elfzoel2 12296 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
215214adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
216213, 215, 2073jca 1234 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
217 elfzle1 12173 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
218217adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
219207zred 11317 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
220214zred 11317 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
221220adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
222193nn0red 11202 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
223222adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
224 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑘𝑖)
225224adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑖)
226 elfzolt2 12306 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
227226adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
228219, 223, 221, 225, 227lelttrd 10047 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
229219, 221, 228ltled 10037 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘𝑁)
230216, 218, 229jca32 555 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
231 elfz2 12162 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
232230, 231sylibr 222 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
233232adantll 745 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
234 dvnmul.dvnf . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
235111a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
236 fvex 6098 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V
237236a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
238235, 237fvmpt2d 6187 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
239238feq1d 5929 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐶𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
240234, 239mpbird 245 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ)
241212, 233, 240syl2anc 690 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘):𝑋⟶ℂ)
2422413adant3 1073 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐶𝑘):𝑋⟶ℂ)
243 simp3 1055 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → 𝑥𝑋)
244242, 243ffvelrnd 6253 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
245193nn0zd 11315 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
246245adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
247246, 207zsubcld 11322 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℤ)
248213, 215, 2473jca 1234 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ))
249 elfzel2 12169 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
250249zred 11317 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
251206zred 11317 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
252250, 251subge0d 10469 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖𝑘) ↔ 𝑘𝑖))
253224, 252mpbird 245 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖𝑘))
254253adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖𝑘))
255223, 219resubcld 10310 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℝ)
256221, 219resubcld 10310 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ∈ ℝ)
257179a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
258221, 257jca 552 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
259 resubcl 10197 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
260258, 259syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
261223, 221, 219, 227ltsub1dd 10491 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁𝑘))
262257, 219, 221, 218lesub2dd 10496 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ (𝑁 − 0))
263255, 256, 260, 261, 262ltletrd 10049 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < (𝑁 − 0))
264220recnd 9925 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
265264subid1d 10233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
266265adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
267263, 266breqtrd 4603 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) < 𝑁)
268255, 221, 267ltled 10037 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ≤ 𝑁)
269248, 254, 268jca32 555 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
270 elfz2 12162 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑖𝑘) ∈ ℤ) ∧ (0 ≤ (𝑖𝑘) ∧ (𝑖𝑘) ≤ 𝑁)))
271269, 270sylibr 222 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
272271adantll 745 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ (0...𝑁))
273 ovex 6555 . . . . . . . . . . . . . . . . . 18 (𝑖𝑘) ∈ V
274 eleq1 2675 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖𝑘) ∈ (0...𝑁)))
275274anbi2d 735 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁))))
276 fveq2 6088 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
277276feq1d 5929 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
278275, 277imbi12d 332 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖𝑘) → (((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)))
279 nfv 1829 . . . . . . . . . . . . . . . . . . 19 𝑘((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
280 eleq1 2675 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
281280anbi2d 735 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑𝑗 ∈ (0...𝑁))))
282 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
283282feq1d 5929 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
284281, 283imbi12d 332 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
285 dvnmul.dvng . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
286279, 284, 285chvar 2249 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
287273, 278, 286vtocl 3231 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
288212, 272, 287syl2anc 690 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ)
289150a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
290 fveq2 6088 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (𝑖𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
291290adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑖𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
292 fvex 6098 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) ∈ V
293292a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) ∈ V)
294289, 291, 271, 293fvmptd 6182 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
295294adantll 745 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
296295feq1d 5929 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)):𝑋⟶ℂ))
297288, 296mpbird 245 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
2982973adant3 1073 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (𝐷‘(𝑖𝑘)):𝑋⟶ℂ)
299298, 243ffvelrnd 6253 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
300244, 299mulcld 9917 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
301211, 300mulcld 9917 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
3022113expa 1256 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (𝑖C𝑘) ∈ ℂ)
303246peano2zd 11320 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
304303, 207zsubcld 11322 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
305213, 215, 3043jca 1234 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
306 peano2re 10061 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
307250, 306syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
308 peano2re 10061 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
309251, 308syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
310251ltp1d 10806 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
311 1red 9912 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
312251, 250, 311, 224leadd1dd 10493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
313251, 309, 307, 310, 312ltletrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
314251, 307, 313ltled 10037 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
315314adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
316223, 306syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
317316, 219subge0d 10469 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
318315, 317mpbird 245 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
319316, 219resubcld 10310 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
320 elfzop1le2 38267 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
321320adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
322316, 221, 219, 321lesub1dd 10495 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁𝑘))
323262, 266breqtrd 4603 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁𝑘) ≤ 𝑁)
324319, 256, 221, 322, 323letrd 10046 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
325305, 318, 324jca32 555 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
326 elfz2 12162 . . . . . . . . . . . . . . . . . . . . 21 (((𝑖 + 1) − 𝑘) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑘) ∧ ((𝑖 + 1) − 𝑘) ≤ 𝑁)))
327325, 326sylibr 222 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
328327adantll 745 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
329 ovex 6555 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) − 𝑘) ∈ V
330 eleq1 2675 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
331330anbi2d 735 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
332 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
333332feq1d 5929 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
334331, 333imbi12d 332 . . . . . . . . . . . . . . . . . . . 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)))
335329, 334, 286vtocl 3231 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
336212, 328, 335syl2anc 690 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
337150a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
338 simpr 475 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
339338fveq2d 6092 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
340 fvex 6098 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V
341340a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
342337, 339, 328, 341fvmptd 6182 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
343342feq1d 5929 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
344336, 343mpbird 245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
345344ffvelrnda 6252 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
3462443expa 1256 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
347345, 346mulcomd 9918 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
348347oveq2d 6543 . . . . . . . . . . . . . 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))))
349207peano2zd 11320 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
350213, 215, 3493jca 1234 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ))
351179a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
352351, 251, 309, 217, 310lelttrd 10047 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
353351, 309, 352ltled 10037 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
354353adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
355219, 308syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
356312adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
357355, 316, 221, 356, 321letrd 10046 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
358350, 354, 357jca32 555 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
359 elfz2 12162 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 + 1) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) ∧ (0 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≤ 𝑁)))
360358, 359sylibr 222 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
361360adantll 745 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
362 ovex 6555 . . . . . . . . . . . . . . . . . . 19 (𝑘 + 1) ∈ V
363 eleq1 2675 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
364363anbi2d 735 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝜑𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
365 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝐶𝑗) = (𝐶‘(𝑘 + 1)))
366365feq1d 5929 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝐶𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
367364, 366imbi12d 332 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 1) → (((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
368 nfv 1829 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝜑𝑗 ∈ (0...𝑁))
369 nfmpt1 4669 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
370111, 369nfcxfr 2748 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝐶
371 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑗
372370, 371nffv 6095 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝐶𝑗)
373 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑋
374 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . 22 𝑘
375372, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶𝑗):𝑋⟶ℂ
376368, 375nfim 1812 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
377 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑗 → (𝐶𝑘) = (𝐶𝑗))
378377feq1d 5929 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶𝑗):𝑋⟶ℂ))
379281, 378imbi12d 332 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)))
380376, 379, 240chvar 2249 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑁)) → (𝐶𝑗):𝑋⟶ℂ)
381362, 367, 380vtocl 3231 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
382212, 361, 381syl2anc 690 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
383382ffvelrnda 6252 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
3842993expa 1256 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖𝑘))‘𝑥) ∈ ℂ)
385383, 384mulcld 9917 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
386345, 346mulcld 9917 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥)) ∈ ℂ)
387385, 386addcld 9916 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶𝑘)‘𝑥))) ∈ ℂ)
388348, 387eqeltrrd 2688 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
389302, 388mulcld 9917 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
3903893impa 1250 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) + (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
391212, 79syl 17 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
392179a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → 0 ∈ ℝ)
393212, 86syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
394391, 393, 210dvmptconst 38627 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ (𝑖C𝑘))) = (𝑥𝑋 ↦ 0))
3953003expa 1256 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
396212, 233, 238syl2anc 690 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
397396eqcomd 2615 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶𝑘))
398241feqmptd 6144 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶𝑘) = (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)))
399397, 398eqtr2d 2644 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
400399oveq2d 6543 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
401391, 80syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
402212, 127syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
403 elfznn0 12260 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
404403adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
405 dvnp1 23439 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
406401, 402, 404, 405syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
407406eqcomd 2615 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
408114a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛)))
409 fveq2 6088 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
410409adantl 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = (𝑘 + 1)) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
411 fvex 6098 . . . . . . . . . . . . . . . . . . 19 ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V
412411a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
413408, 410, 361, 412fvmptd 6182 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
414413eqcomd 2615 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
415382feqmptd 6144 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
416414, 415eqtrd 2643 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
417400, 407, 4163eqtrd 2647 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐶𝑘)‘𝑥))) = (𝑥𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
418295eqcomd 2615 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)) = (𝐷‘(𝑖𝑘)))
419297feqmptd 6144 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖𝑘)) = (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)))
420418, 419eqtr2d 2644 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖𝑘)))
421420oveq2d 6543 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
422212, 158syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
423 fznn0sub 12202 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → (𝑖𝑘) ∈ ℕ0)
424423adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖𝑘) ∈ ℕ0)
425 dvnp1 23439 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
426401, 422, 424, 425syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))))
427426eqcomd 2615 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)))
428223recnd 9925 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
429 1cnd 9913 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
430219recnd 9925 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
431428, 429, 430addsubd 10265 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖𝑘) + 1))
432431eqcomd 2615 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖𝑘) + 1) = ((𝑖 + 1) − 𝑘))
433432fveq2d 6092 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
434433adantll 745 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
435342eqcomd 2615 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
436344feqmptd 6144 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
437434, 435, 4363eqtrd 2647 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖𝑘) + 1)) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
438421, 427, 4373eqtrd 2647 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥𝑋 ↦ ((𝐷‘(𝑖𝑘))‘𝑥))) = (𝑥𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
439391, 346, 383, 417, 384, 345, 438dvmptmul 23475 . . . . . . . . . . . . 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)))))
440391, 302, 392, 394, 395, 387, 439dvmptmul 23475 . . . . . . . . . . . 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)))))
441395mul02d 10086 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → (0 · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = 0)
442348oveq1d 6542 . . . . . . . . . . . . . . . 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)))
443388, 302mulcomd 9918 . . . . . . . . . . . . . . . 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)))))
444442, 443eqtrd 2643 . . . . . . . . . . . . . . 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)))))
445441, 444oveq12d 6545 . . . . . . . . . . . . . 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))))))
446389addid2d 10089 . . . . . . . . . . . . . 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)))))
447445, 446eqtrd 2643 . . . . . . . . . . . . 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)))))
448447mpteq2dva 4666 . . . . . . . . . . . 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))))))
449440, 448eqtrd 2643 . . . . . . . . . . 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))))))
450200, 201, 202, 203, 204, 301, 390, 449dvmptfsum 23487 . . . . . . . . . 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))))))
451210adantlr 746 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
452385an32s 841 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) ∈ ℂ)
453 anass 678 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)))
454 ancom 464 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋) ↔ (𝑥𝑋𝑘 ∈ (0...𝑖)))
455454anbi2i 725 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
456 anass 678 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))))
457456bicomi 212 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑥𝑋𝑘 ∈ (0...𝑖))) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
458455, 457bitri 262 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥𝑋)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
459453, 458bitri 262 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)))
460459imbi1i 337 . . . . . . . . . . . . . . . 16 (((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥𝑋) → ((𝐶𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ))
461346, 460mpbi 218 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
462459imbi1i 337 . . . . . . . . . . . . . . . 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))
463345, 462mpbi 218 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
464461, 463mulcld 9917 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
465451, 452, 464adddid 9921 . . . . . . . . . . . . 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)))))
466465sumeq2dv 14230 . . . . . . . . . . . 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)))))
467204adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...𝑖) ∈ Fin)
468451, 452mulcld 9917 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ)
469451, 464mulcld 9917 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
470467, 468, 469fsumadd 14266 . . . . . . . . . . . 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)))))
471 oveq2 6535 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (𝑖C𝑘) = (𝑖C))
472 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑘 + 1) = ( + 1))
473472fveq2d 6092 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐶‘(𝑘 + 1)) = (𝐶‘( + 1)))
474473fveq1d 6090 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘( + 1))‘𝑥))
475 oveq2 6535 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = → (𝑖𝑘) = (𝑖))
476475fveq2d 6092 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = → (𝐷‘(𝑖𝑘)) = (𝐷‘(𝑖)))
477476fveq1d 6090 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → ((𝐷‘(𝑖𝑘))‘𝑥) = ((𝐷‘(𝑖))‘𝑥))
478474, 477oveq12d 6545 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)) = (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
479471, 478oveq12d 6545 . . . . . . . . . . . . . . . . . 18 (𝑘 = → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))))
480 nfcv 2750 . . . . . . . . . . . . . . . . . 18 (0...𝑖)
481 nfcv 2750 . . . . . . . . . . . . . . . . . 18 𝑘(0...𝑖)
482 nfcv 2750 . . . . . . . . . . . . . . . . . 18 ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥)))
483 nfcv 2750 . . . . . . . . . . . . . . . . . . 19 𝑘(𝑖C)
484 nfcv 2750 . . . . . . . . . . . . . . . . . . 19 𝑘 ·
485 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . 22 𝑘( + 1)
486370, 485nffv 6095 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐶‘( + 1))
487 nfcv 2750 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑥
488486, 487nffv 6095 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐶‘( + 1))‘𝑥)
489 nfmpt1 4669 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
490147, 489nfcxfr 2748 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝐷
491 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(𝑖)
492490, 491nffv 6095 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝐷‘(𝑖))
493492, 487nffv 6095 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝐷‘(𝑖))‘𝑥)
494488, 484, 493nfov 6553 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))
495483, 484, 494nfov 6553 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
496479, 480, 481, 482, 495cbvsum 14222 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) = Σ ∈ (0...𝑖)((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)))
497496a1i 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))))
498 1zzd 11244 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 1 ∈ ℤ)
49995a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ ℤ)
500245ad2antlr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑖 ∈ ℤ)
501 nfv 1829 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
502 nfcv 2750 . . . . . . . . . . . . . . . . . . . . 21 𝑘
503502, 481nfel 2762 . . . . . . . . . . . . . . . . . . . 20 𝑘 ∈ (0...𝑖)
504501, 503nfan 1815 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))
505495, 374nfel 2762 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ
506504, 505nfim 1812 . . . . . . . . . . . . . . . . . 18 𝑘((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
507 eleq1 2675 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = → (𝑘 ∈ (0...𝑖) ↔ ∈ (0...𝑖)))
508507anbi2d 735 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖))))
509479eleq1d 2671 . . . . . . . . . . . . . . . . . . 19 (𝑘 = → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ))
510508, 509imbi12d 332 . . . . . . . . . . . . . . . . . 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)))
511506, 510, 468chvar 2249 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ ∈ (0...𝑖)) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) ∈ ℂ)
512 oveq2 6535 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (𝑖C) = (𝑖C(𝑗 − 1)))
513 oveq1 6534 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → ( + 1) = ((𝑗 − 1) + 1))
514513fveq2d 6092 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐶‘( + 1)) = (𝐶‘((𝑗 − 1) + 1)))
515514fveq1d 6090 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐶‘( + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
516 oveq2 6535 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑗 − 1) → (𝑖) = (𝑖 − (𝑗 − 1)))
517516fveq2d 6092 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑗 − 1) → (𝐷‘(𝑖)) = (𝐷‘(𝑖 − (𝑗 − 1))))
518517fveq1d 6090 . . . . . . . . . . . . . . . . . . 19 ( = (𝑗 − 1) → ((𝐷‘(𝑖))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
519515, 518oveq12d 6545 . . . . . . . . . . . . . . . . . 18 ( = (𝑗 − 1) → (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
520512, 519oveq12d 6545 . . . . . . . . . . . . . . . . 17 ( = (𝑗 − 1) → ((𝑖C) · (((𝐶‘( + 1))‘𝑥) · ((𝐷‘(𝑖))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
521498, 499, 500, 511, 520fsumshft 14303 . . . . . . . . . . . . . . . 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))))
522497, 521eqtrd 2643 . . . . . . . . . . . . . . 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))))
523 0p1e1 10982 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
524523oveq1i 6537 . . . . . . . . . . . . . . . . 17 ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
525524sumeq1i 14225 . . . . . . . . . . . . . . . 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)))
526525a1i 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))))
527 elfzelz 12171 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
528527zcnd 11318 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
529 1cnd 9913 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
530528, 529npcand 10248 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
531530fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶𝑗))
532531fveq1d 6090 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
533532adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶𝑗)‘𝑥))
534222recnd 9925 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
535534adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
536528adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
537529adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
538535, 536, 537subsub3d 10274 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
539538fveq2d 6092 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
540539fveq1d 6090 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
541533, 540oveq12d 6545 . . . . . . . . . . . . . . . . . . 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)))
542541oveq2d 6543 . . . . . . . . . . . . . . . . . 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))))
543542sumeq2dv 14230 . . . . . . . . . . . . . . . . 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))))
544543ad2antlr 758 . . . . . . . . . . . . . . . 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))))
545 nfv 1829 . . . . . . . . . . . . . . . . 17 𝑗((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋)
546 nfcv 2750 . . . . . . . . . . . . . . . . 17 𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
547 fzfid 12592 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...(𝑖 + 1)) ∈ Fin)
548193adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
549527adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
550 1zzd 11244 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
551549, 550zsubcld 11322 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
552548, 551bccld 38296 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
553552nn0cnd 11203 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
554553adantll 745 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
555554adantlr 746 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
5561ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
557 0zd 11225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
558214adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
559557, 558, 5493jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ))
560179a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
561527zred 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
562 1red 9912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
563 0lt1 10402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 < 1
564563a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
565 elfzle1 12173 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
566560, 562, 561, 564, 565ltletrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
567560, 561, 566ltled 10037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
568567adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
569561adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
570222adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
571 1red 9912 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
572570, 571readdcld 9926 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
573220adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
574 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
575574adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
576320adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
577569, 572, 573, 575, 576letrd 10046 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗𝑁)
578559, 568, 577jca32 555 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
579 elfz2 12162 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑁)))
580578, 579sylibr 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
581580adantll 745 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
582556, 581, 380syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
583582adantlr 746 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶𝑗):𝑋⟶ℂ)
584 simplr 787 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥𝑋)
585583, 584ffvelrnd 6253 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
586245adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
587586peano2zd 11320 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
588587, 549zsubcld 11322 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
589557, 558, 5883jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ))
590572, 569subge0d 10469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
591575, 590mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
592572, 569resubcld 10310 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
593592leidd 10446 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
594561, 566elrpd 11704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
595594adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
596572, 595ltsubrpd 11739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
597592, 572, 573, 596, 576ltletrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
598592, 592, 573, 593, 597lelttrd 10047 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
599592, 573, 598ltled 10037 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
600589, 591, 599jca32 555 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
601 elfz2 12162 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 + 1) − 𝑗) ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑗) ∈ ℤ) ∧ (0 ≤ ((𝑖 + 1) − 𝑗) ∧ ((𝑖 + 1) − 𝑗) ≤ 𝑁)))
602600, 601sylibr 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
603602adantll 745 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
604 nfv 1829 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
605 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘((𝑖 + 1) − 𝑗)
606490, 605nffv 6095 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘((𝑖 + 1) − 𝑗))
607606, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
608604, 607nfim 1812 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
609 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 + 1) − 𝑗) ∈ V
610 eleq1 2675 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
611610anbi2d 735 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
612 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
613612feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
614611, 613imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
615147a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
616 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V
617616a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
618615, 617fvmpt2d 6187 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
619618feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝐷𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
620285, 619mpbird 245 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ)
621608, 609, 614, 620vtoclf 3230 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
622556, 603, 621syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
623622adantlr 746 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
624623, 584ffvelrnd 6253 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
625585, 624mulcld 9917 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
626555, 625mulcld 9917 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
627 1zzd 11244 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
628245peano2zd 11320 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
629523eqcomi 2618 . . . . . . . . . . . . . . . . . . . . . . 23 1 = (0 + 1)
630629a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
631179a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
632 1red 9912 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
633193nn0ge0d 11204 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
634631, 222, 632, 633leadd1dd 10493 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
635630, 634eqbrtrd 4599 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
636627, 628, 6353jca 1234 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
637 eluz2 11528 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
638636, 637sylibr 222 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ‘1))
639 eluzfz2 12178 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
640638, 639syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
641640ad2antlr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
642 oveq1 6534 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
643642oveq2d 6543 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
644 fveq2 6088 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐶𝑗) = (𝐶‘(𝑖 + 1)))
645644fveq1d 6090 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐶𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
646 oveq2 6535 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
647646fveq2d 6092 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
648647fveq1d 6090 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
649645, 648oveq12d 6545 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
650643, 649oveq12d 6545 . . . . . . . . . . . . . . . . 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))))
651545, 546, 547, 626, 641, 650fsumsplit1 38463 . . . . . . . . . . . . . . . 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)))))
652 1cnd 9913 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
653534, 652pncand 10245 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
654653oveq2d 6543 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
655 bcnn 12919 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
656193, 655syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
657654, 656eqtrd 2643 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
658534, 652addcld 9916 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
659658subidd 10232 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
660659fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
661660fveq1d 6090 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
662661oveq2d 6543 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
663657, 662oveq12d 6545 . . . . . . . . . . . . . . . . . . 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))))
664663ad2antlr 758 . . . . . . . . . . . . . . . . . 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))))
665 simpl 471 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝜑)
666 fzofzp1 12389 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
667666adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
668 nfv 1829 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
669 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝑖 + 1)
670370, 669nffv 6095 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐶‘(𝑖 + 1))
671670, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
672668, 671nfim 1812 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
673 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 + 1) ∈ V
674 eleq1 2675 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
675674anbi2d 735 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
676 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝐶𝑘) = (𝐶‘(𝑖 + 1)))
677676feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
678675, 677imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
679672, 673, 678, 240vtoclf 3230 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
680665, 667, 679syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
681680ffvelrnda 6252 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
682 nfv 1829 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
683 nfcv 2750 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑘0
684490, 683nffv 6095 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑘(𝐷‘0)
685684, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑘(𝐷‘0):𝑋⟶ℂ
686682, 685nfim 1812 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
687 c0ex 9891 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
688 eleq1 2675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
689688anbi2d 735 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝜑𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
690 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝐷𝑘) = (𝐷‘0))
691690feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
692689, 691imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
693686, 687, 692, 620vtoclf 3230 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
6941, 119, 693syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐷‘0):𝑋⟶ℂ)
695694adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
696695ffvelrnda 6252 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
697681, 696mulcld 9917 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
698697mulid2d 9915 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
699664, 698eqtrd 2643 . . . . . . . . . . . . . . . . 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)))
700 1m1e0 10939 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 − 1) = 0
701700fveq2i 6091 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘(1 − 1)) = (ℤ‘0)
7023eqcomi 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ‘0) = ℕ0
703701, 702eqtr2i 2632 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘(1 − 1))
704703a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ‘(1 − 1)))
705193, 704eleqtrd 2689 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘(1 − 1)))
706 fzdifsuc2 38290 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (ℤ‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
707705, 706syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
708707eqcomd 2615 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
709708sumeq1d 14228 . . . . . . . . . . . . . . . . . 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))))
710709ad2antlr 758 . . . . . . . . . . . . . . . . 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))))
711699, 710oveq12d 6545 . . . . . . . . . . . . . . . 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)))))
712544, 651, 7113eqtrd 2647 . . . . . . . . . . . . . . 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)))))
713522, 526, 7123eqtrd 2647 . . . . . . . . . . . . . 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)))))
714 nfcv 2750 . . . . . . . . . . . . . . . . 17 𝑘(𝑖C0)
715370, 683nffv 6095 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐶‘0)
716715, 487nffv 6095 . . . . . . . . . . . . . . . . . 18 𝑘((𝐶‘0)‘𝑥)
717 nfcv 2750 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1) − 0)
718490, 717nffv 6095 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐷‘((𝑖 + 1) − 0))
719718, 487nffv 6095 . . . . . . . . . . . . . . . . . 18 𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
720716, 484, 719nfov 6553 . . . . . . . . . . . . . . . . 17 𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
721714, 484, 720nfov 6553 . . . . . . . . . . . . . . . 16 𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
722702a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (ℤ‘0) = ℕ0)
723193, 722eleqtrrd 2690 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ‘0))
724 eluzfz1 12177 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ‘0) → 0 ∈ (0...𝑖))
725723, 724syl 17 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
726725ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 0 ∈ (0...𝑖))
727 oveq2 6535 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
728109fveq1d 6090 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐶𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
729 oveq2 6535 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
730729fveq2d 6092 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
731730fveq1d 6090 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
732728, 731oveq12d 6545 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
733727, 732oveq12d 6545 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
734501, 721, 467, 469, 726, 733fsumsplit1 38463 . . . . . . . . . . . . . . 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)))))
735658subid1d 10233 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
736735fveq2d 6092 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
737736fveq1d 6090 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
738737oveq2d 6543 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
739738oveq2d 6543 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
740739oveq1d 6542 . . . . . . . . . . . . . . . 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)))))
741740ad2antlr 758 . . . . . . . . . . . . . . 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)))))
742 bcn0 12917 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
743193, 742syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
744743oveq1d 6542 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
745744ad2antlr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
746715, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐶‘0):𝑋⟶ℂ
747682, 746nfim 1812 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
748109feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → ((𝐶𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
749689, 748imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐶𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
750747, 687, 749, 240vtoclf 3230 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
7511, 119, 750syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐶‘0):𝑋⟶ℂ)
752751adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
753752ffvelrnda 6252 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
754490, 669nffv 6095 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘(𝐷‘(𝑖 + 1))
755754, 373, 374nff 5940 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
756668, 755nfim 1812 . . . . . . . . . . . . . . . . . . . . . 22 𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
757 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → (𝐷𝑘) = (𝐷‘(𝑖 + 1)))
758757feq1d 5929 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → ((𝐷𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
759675, 758imbi12d 332 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑖 + 1) → (((𝜑𝑘 ∈ (0...𝑁)) → (𝐷𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
760756, 673, 759, 620vtoclf 3230 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
761665, 667, 760syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
762761ffvelrnda 6252 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
763753, 762mulcld 9917 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
764763mulid2d 9915 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
765745, 764eqtrd 2643 . . . . . . . . . . . . . . . 16 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
766 nfv 1829 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑖 ∈ (0..^𝑁)
767 1zzd 11244 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
768245adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
769 eldifi 3693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
770 elfzelz 12171 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
771769, 770syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
772771adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
773767, 768, 7723jca 1234 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → (1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
774 elfznn0 12260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
775769, 774syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
776 eldifsni 4260 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
777775, 776jca 552 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0𝑗 ≠ 0))
778 elnnne0 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0𝑗 ≠ 0))
779777, 778sylibr 222 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
780 nnge1 10896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ → 1 ≤ 𝑗)
781779, 780syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
782781adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
783 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑖) → 𝑗𝑖)
784769, 783syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗𝑖)
785784adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗𝑖)
786773, 782, 785jca32 555 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
787 elfz2 12162 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) ↔ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (1 ≤ 𝑗𝑗𝑖)))
788786, 787sylibr 222 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
789788ex 448 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
790 0zd 11225 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
791 elfzel2 12169 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
792 elfzelz 12171 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
793790, 791, 7923jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → (0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
794179a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
795792zred 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
796 1red 9912 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
797563a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 0 < 1)
798 elfzle1 12173 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
799794, 796, 795, 797, 798ltletrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
800794, 795, 799ltled 10037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
801 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 𝑗𝑖)
802793, 800, 801jca32 555 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
803 elfz2 12162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) ↔ ((0 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ≤ 𝑗𝑗𝑖)))
804802, 803sylibr 222 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
805794, 799gtned 10024 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
806 nelsn 4158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
807805, 806syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
808804, 807eldifd 3550 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
809808adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
810809ex 448 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
811789, 810impbid 200 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
812766, 811alrimi 2068 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
813 dfcleq 2603 . . . . . . . . . . . . . . . . . . 19 (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
814812, 813sylibr 222 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
815814sumeq1d 14228 . . . . . . . . . . . . . . . . 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))))
816815ad2antlr 758 . . . . . . . . . . . . . . . 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))))
817765, 816oveq12d 6545 . . . . . . . . . . . . . . 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)))))
818734, 741, 8173eqtrd 2647 . . . . . . . . . . . . . 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)))))
819713, 818oveq12d 6545 . . . . . . . . . . . . 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))))))
820 fzfid 12592 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) ∈ Fin)
821193adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
822809, 771syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
823 1zzd 11244 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
824822, 823zsubcld 11322 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
825821, 824bccld 38296 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
826825nn0cnd 11203 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
827826adantll 745 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
828827adantlr 746 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
829 simpl 471 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋))
830 fzelp1 12221 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
831830adantl 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
832829, 831, 585syl2anc 690 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶𝑗)‘𝑥) ∈ ℂ)
833831, 624syldan 485 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
834832, 833mulcld 9917 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
835828, 834mulcld 9917 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
836820, 835fsumcl 14260 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
837193adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
838 elfzelz 12171 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
839838adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
840837, 839bccld 38296 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
841840nn0cnd 11203 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
842841adantll 745 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
843842adantlr 746 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
844 simpll 785 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑𝑖 ∈ (0..^𝑁)))
845 simplr 787 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥𝑋)
846804ssriv 3571 . . . . . . . . . . . . . . . . . . . 20 (1...𝑖) ⊆ (0...𝑖)
847 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
848846, 847sseldi 3565 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
849848adantl 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
850844, 845, 849, 461syl21anc 1316 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
851849, 463syldan 485 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
852850, 851mulcld 9917 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
853843, 852mulcld 9917 . . . . . . . . . . . . . . 15 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
854820, 853fsumcl 14260 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
855697, 836, 763, 854add4d 10116 . . . . . . . . . . . . 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))))))
856 oveq1 6534 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
857856oveq2d 6543 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
858 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐶𝑗) = (𝐶𝑘))
859858fveq1d 6090 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐶𝑗)‘𝑥) = ((𝐶𝑘)‘𝑥))
860 oveq2 6535 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
861860fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
862861fveq1d 6090 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
863859, 862oveq12d 6545 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
864857, 863oveq12d 6545 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
865 nfcv 2750 . . . . . . . . . . . . . . . . . . 19 𝑘(1...𝑖)
866 nfcv 2750 . . . . . . . . . . . . . . . . . . 19 𝑗(1...𝑖)
867 nfcv 2750 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝑖C(𝑗 − 1))
868372, 487nffv 6095 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐶𝑗)‘𝑥)
869606, 487nffv 6095 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
870868, 484, 869nfov 6553 . . . . . . . . . . . . . . . . . . . 20 𝑘(((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
871867, 484, 870nfov 6553 . . . . . . . . . . . . . . . . . . 19 𝑘((𝑖C(𝑗 − 1)) · (((𝐶𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
872 nfcv 2750 . . . . . . . . . . . . . . . . . . 19 𝑗((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
873864, 865, 866, 871, 872cbvsum 14222 . . . . . . . . . . . . . . . . . 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)))
874873a1i 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))))
875874oveq1d 6542 . . . . . . . . . . . . . . . 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)))))
876 peano2zm 11256 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
877839, 876syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
878837, 877bccld 38296 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
879878nn0cnd 11203 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
880879adantll 745 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
881880adantlr 746 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
882881, 852mulcld 9917 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
883820, 882, 853fsumadd 14266 . . . . . . . . . . . . . . . . 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)))))
884883eqcomd 2615 . . . . . . . . . . . . . . . 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)))))
885879, 841addcomd 10090 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
886 bcpasc 12928 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ ℕ0𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
887837, 839, 886syl2anc 690 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
888885, 887eqtr2d 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
889888oveq1d 6542 . . . . . . . . . . . . . . . . . . . 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))))
890889adantll 745 . . . . . . . . . . . . . . . . . . 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))))
891890adantlr 746 . . . . . . . . . . . . . . . . . 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))))
892881, 843, 852adddird 9922 . . . . . . . . . . . . . . . . . 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)))))
893891, 892eqtr2d 2644 . . . . . . . . . . . . . . . . 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))))
894893sumeq2dv 14230 . . . . . . . . . . . . . . . 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))))
895875, 884, 8943eqtrd 2647 . . . . . . . . . . . . . . 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))))
896895oveq2d 6543 . . . . . . . . . . . . . 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)))))
897 peano2nn0 11183 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
898837, 897syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
899898, 839bccld 38296 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
900899nn0cnd 11203 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
901900adantll 745 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
902901adantlr 746 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
903902, 852mulcld 9917 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
904820, 903fsumcl 14260 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
905697, 763, 904addassd 9919 . . . . . . . . . . . . . 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))))))
906193, 897syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
907 bcn0 12917 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
908906, 907syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
909908, 738oveq12d 6545 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
910909ad2antlr 758 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
911910, 764eqtr2d 2644 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
912814ad2antlr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
913912eqcomd 2615 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
914913sumeq1d 14228 . . . . . . . . . . . . . . . . . 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))))
915911, 914oveq12d 6545 . . . . . . . . . . . . . . . . 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)))))
916 nfcv 2750 . . . . . . . . . . . . . . . . . . . 20 𝑘((𝑖 + 1)C0)
917916, 484, 720nfov 6553 . . . . . . . . . . . . . . . . . . 19 𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
918205, 897syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
919918, 207bccld 38296 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
920919nn0cnd 11203 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
921920adantll 745 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
922921adantlr 746 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
923922, 464mulcld 9917 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
924 oveq2 6535 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
925924, 732oveq12d 6545 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
926501, 917, 467, 923, 726, 925fsumsplit1 38463 . . . . . . . . . . . . . . . . . 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)))))
927926eqcomd 2615 . . . . . . . . . . . . . . . . 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))))
928915, 927eqtrd 2643 . . . . . . . . . . . . . . . 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))))
929928oveq2d 6543 . . . . . . . . . . . . . . 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)))))
930 bcnn 12919 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
931906, 930syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
932931ad2antlr 758 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
933932oveq1d 6542 . . . . . . . . . . . . . . . . 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))))
934660adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
935934feq1d 5929 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
936695, 935mpbird 245 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
937936adantr 479 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
938 simpr 475 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → 𝑥𝑋)
939937, 938ffvelrnd 6253 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
940681, 939mulcld 9917 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
941940mulid2d 9915 . . . . . . . . . . . . . . . . 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)))
942662ad2antlr 758 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
943933, 941, 9423eqtrrd 2648 . . . . . . . . . . . . . . . 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))))
944 fzdifsuc 12228 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
945723, 944syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
946945sumeq1d 14228 . . . . . . . . . . . . . . . . 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))))
947946ad2antlr 758 . . . . . . . . . . . . . . . 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))))
948943, 947oveq12d 6545 . . . . . . . . . . . . . . 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)))))
949 nfcv 2750 . . . . . . . . . . . . . . . . . 18 𝑘((𝑖 + 1)C(𝑖 + 1))
950670, 487nffv 6095 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐶‘(𝑖 + 1))‘𝑥)
951 nfcv 2750 . . . . . . . . . . . . . . . . . . . . 21 𝑘((𝑖 + 1) − (𝑖 + 1))
952490, 951nffv 6095 . . . . . . . . . . . . . . . . . . . 20 𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
953952, 487nffv 6095 . . . . . . . . . . . . . . . . . . 19 𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
954950, 484, 953nfov 6553 . . . . . . . . . . . . . . . . . 18 𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
955949, 484, 954nfov 6553 . . . . . . . . . . . . . . . . 17 𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
956 fzfid 12592 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) → (0...(𝑖 + 1)) ∈ Fin)
957906adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
958 elfzelz 12171 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
959958adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
960957, 959bccld 38296 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
961960nn0cnd 11203 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
962961adantll 745 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
963962adantlr 746 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
964665adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
96595a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
966214adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
967965, 966, 9593jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
968 elfzle1 12173 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
969968adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
970959zred 11317 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
971957nn0red 11202 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
972220adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
973 elfzle2 12174 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
974973adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
975320adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
976970, 971, 972, 974, 975letrd 10046 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘𝑁)
977967, 969, 976jca32 555 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘𝑘𝑁)))
978977, 231sylibr 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
979978adantll 745 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
980964, 979, 240syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
981980adantlr 746 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶𝑘):𝑋⟶ℂ)
982 simplr 787 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥𝑋)
983981, 982ffvelrnd 6253 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶𝑘)‘𝑥) ∈ ℂ)
984964adantlr 746 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑥𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
985628adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
986985, 959zsubcld 11322 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
987965, 966, 9863jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ((𝑖 + 1) − 𝑘) ∈ ℤ))
988971, 970subge0d 10469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
989974