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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvnmul Structured version   Visualization version   GIF version
Theorem dvnmul 45331
Description: Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
dvnmul.s (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnmul.x (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
dvnmul.a ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvnmul.cc ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
dvnmul.n (𝜑 → 𝑁 ∈ ℕ0)
dvnmulf 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴)
dvnmul.f 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)
dvnmul.dvnf ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
dvnmul.dvng ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
dvnmul.c 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
dvnmul.d 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
Assertion
Ref Expression
dvnmul (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑥,𝐶   𝑥,𝐷   𝑘,𝐹   𝑘,𝐺   𝑘,𝑁,𝑥   𝑆,𝑘,𝑥   𝑘,𝑋,𝑥   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑘)   𝐷(𝑘)   𝐹(𝑥)   𝐺(𝑥)
Proof of Theorem dvnmul
Dummy variables 𝑖 𝑚 𝑛 ℎ 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝜑 → 𝜑)
2 dvnmul.n . . 3 (𝜑 → 𝑁 ∈ ℕ0)
3 nn0uz 12895 . . . . 5 ℕ0 = (ℤ≥‘0)
42, 3eleqtrdi 2839 . . . 4 (𝜑 → 𝑁 ∈ (ℤ≥‘0))
5 eluzfz2 13542 . . . 4 (𝑁 ∈ (ℤ≥‘0) → 𝑁 ∈ (0...𝑁))
64, 5syl 17 . . 3 (𝜑 → 𝑁 ∈ (0...𝑁))
7 eleq1 2817 . . . . 5 (𝑛 = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑁 ∈ (0...𝑁)))
8 fveq2 6897 . . . . . . 7 (𝑛 = 𝑁 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁))
9 oveq2 7428 . . . . . . . . . 10 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
109sumeq1d 15680 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
11 oveq1 7427 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
12 fvoveq1 7443 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝐷‘(𝑛 − 𝑘)) = (𝐷‘(𝑁 − 𝑘)))
1312fveq1d 6899 . . . . . . . . . . . 12 (𝑛 = 𝑁 → ((𝐷‘(𝑛 − 𝑘))‘𝑥) = ((𝐷‘(𝑁 − 𝑘))‘𝑥))
1413oveq2d 7436 . . . . . . . . . . 11 (𝑛 = 𝑁 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))
1511, 14oveq12d 7438 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = ((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
1615sumeq2sdv 15683 . . . . . . . . 9 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑁)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
1710, 16eqtrd 2768 . . . . . . . 8 (𝑛 = 𝑁 → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))
1817mpteq2dv 5250 . . . . . . 7 (𝑛 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))
198, 18eqeq12d 2744 . . . . . 6 (𝑛 = 𝑁 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))
2019imbi2d 340 . . . . 5 (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))
217, 20imbi12d 344 . . . 4 (𝑛 = 𝑁 → ((𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))) ↔ (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))))
22 fveq2 6897 . . . . . . 7 (𝑚 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0))
23 simpl 482 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 0)
2423oveq2d 7436 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...0))
25 simpll 766 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → 𝑚 = 0)
2625oveq1d 7435 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝑚C𝑘) = (0C𝑘))
2725fvoveq1d 7442 . . . . . . . . . . . 12 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(0 − 𝑘)))
2827fveq1d 6899 . . . . . . . . . . 11 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(0 − 𝑘))‘𝑥))
2928oveq2d 7436 . . . . . . . . . 10 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
3026, 29oveq12d 7438 . . . . . . . . 9 (((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...0)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3124, 30sumeq12rdv 15686 . . . . . . . 8 ((𝑚 = 0 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
3231mpteq2dva 5248 . . . . . . 7 (𝑚 = 0 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
3322, 32eqeq12d 2744 . . . . . 6 (𝑚 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
3433imbi2d 340 . . . . 5 (𝑚 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))))
35 fveq2 6897 . . . . . . 7 (𝑚 = 𝑖 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖))
36 simpl 482 . . . . . . . . . 10 ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑖)
3736oveq2d 7436 . . . . . . . . 9 ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑖))
38 simpll 766 . . . . . . . . . . 11 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → 𝑚 = 𝑖)
3938oveq1d 7435 . . . . . . . . . 10 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑚C𝑘) = (𝑖C𝑘))
4038fvoveq1d 7442 . . . . . . . . . . . 12 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑖 − 𝑘)))
4140fveq1d 6899 . . . . . . . . . . 11 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − 𝑘))‘𝑥))
4241oveq2d 7436 . . . . . . . . . 10 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
4339, 42oveq12d 7438 . . . . . . . . 9 (((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))
4437, 43sumeq12rdv 15686 . . . . . . . 8 ((𝑚 = 𝑖 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))
4544mpteq2dva 5248 . . . . . . 7 (𝑚 = 𝑖 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
4635, 45eqeq12d 2744 . . . . . 6 (𝑚 = 𝑖 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
4746imbi2d 340 . . . . 5 (𝑚 = 𝑖 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))))
48 fveq2 6897 . . . . . . 7 (𝑚 = (𝑖 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)))
49 simpl 482 . . . . . . . . . 10 ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → 𝑚 = (𝑖 + 1))
5049oveq2d 7436 . . . . . . . . 9 ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...(𝑖 + 1)))
51 simpll 766 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑚 = (𝑖 + 1))
5251oveq1d 7435 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑚C𝑘) = ((𝑖 + 1)C𝑘))
5351fvoveq1d 7442 . . . . . . . . . . . 12 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
5453fveq1d 6899 . . . . . . . . . . 11 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
5554oveq2d 7436 . . . . . . . . . 10 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
5652, 55oveq12d 7438 . . . . . . . . 9 (((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
5750, 56sumeq12rdv 15686 . . . . . . . 8 ((𝑚 = (𝑖 + 1) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
5857mpteq2dva 5248 . . . . . . 7 (𝑚 = (𝑖 + 1) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
5948, 58eqeq12d 2744 . . . . . 6 (𝑚 = (𝑖 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
6059imbi2d 340 . . . . 5 (𝑚 = (𝑖 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
61 fveq2 6897 . . . . . . 7 (𝑚 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛))
62 simpl 482 . . . . . . . . . 10 ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → 𝑚 = 𝑛)
6362oveq2d 7436 . . . . . . . . 9 ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → (0...𝑚) = (0...𝑛))
64 simpll 766 . . . . . . . . . . 11 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → 𝑚 = 𝑛)
6564oveq1d 7435 . . . . . . . . . 10 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝑚C𝑘) = (𝑛C𝑘))
6664fvoveq1d 7442 . . . . . . . . . . . 12 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (𝐷‘(𝑚 − 𝑘)) = (𝐷‘(𝑛 − 𝑘)))
6766fveq1d 6899 . . . . . . . . . . 11 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐷‘(𝑚 − 𝑘))‘𝑥) = ((𝐷‘(𝑛 − 𝑘))‘𝑥))
6867oveq2d 7436 . . . . . . . . . 10 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))
6965, 68oveq12d 7438 . . . . . . . . 9 (((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = ((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
7063, 69sumeq12rdv 15686 . . . . . . . 8 ((𝑚 = 𝑛 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))
7170mpteq2dva 5248 . . . . . . 7 (𝑚 = 𝑛 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))
7261, 71eqeq12d 2744 . . . . . 6 (𝑚 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥)))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))
7372imbi2d 340 . . . . 5 (𝑚 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑚) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑚 − 𝑘))‘𝑥))))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥)))))))
74 dvnmul.s . . . . . . . . 9 (𝜑 → 𝑆 ∈ {ℝ, ℂ})
75 recnprss 25846 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
7674, 75syl 17 . . . . . . . 8 (𝜑 → 𝑆 ⊆ ℂ)
77 dvnmul.a . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
78 dvnmul.cc . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
7977, 78mulcld 11265 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) ∈ ℂ)
80 restsspw 17413 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆
81 dvnmul.x . . . . . . . . . . 11 (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
8280, 81sselid 3978 . . . . . . . . . 10 (𝜑 → 𝑋 ∈ 𝒫 𝑆)
83 elpwi 4610 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
8482, 83syl 17 . . . . . . . . 9 (𝜑 → 𝑋 ⊆ 𝑆)
85 cnex 11220 . . . . . . . . . 10 ℂ ∈ V
8685a1i 11 . . . . . . . . 9 (𝜑 → ℂ ∈ V)
8779, 84, 86, 74mptelpm 44549 . . . . . . . 8 (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
88 dvn0 25867 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
8976, 87, 88syl2anc 583 . . . . . . 7 (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
90 0z 12600 . . . . . . . . . . . 12 0 ∈ ℤ
91 fzsn 13576 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
9290, 91ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
9392sumeq1i 15677 . . . . . . . . . 10 Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))
9493a1i 11 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
95 nfcvd 2900 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ 𝑋) → Ⅎ𝑘(𝐴 · 𝐵))
96 nfv 1910 . . . . . . . . . 10 Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
97 oveq2 7428 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C𝑘) = (0C0))
98 0nn0 12518 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
99 bcn0 14302 . . . . . . . . . . . . . . . 16 (0 ∈ ℕ0 → (0C0) = 1)
10098, 99ax-mp 5 . . . . . . . . . . . . . . 15 (0C0) = 1
101100a1i 11 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0C0) = 1)
10297, 101eqtrd 2768 . . . . . . . . . . . . 13 (𝑘 = 0 → (0C𝑘) = 1)
103102adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (0C𝑘) = 1)
104 fveq2 6897 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐶‘𝑘) = (𝐶‘0))
105104adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = (𝐶‘0))
106 dvnmul.c . . . . . . . . . . . . . . . . . . . . 21 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
107 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑛))
108107cbvmptv 5261 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
109106, 108eqtri 2756 . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑛))
110 fveq2 6897 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘0))
111 eluzfz1 13541 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑁))
1124, 111syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 0 ∈ (0...𝑁))
113 fvexd 6912 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) ∈ V)
114109, 110, 112, 113fvmptd3 7028 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
115114adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 = 0) → (𝐶‘0) = ((𝑆 D𝑛 𝐹)‘0))
116105, 115eqtrd 2768 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘0))
117 dvnmulf . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴)
11877, 84, 86, 74mptelpm 44549 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
119117, 118eqeltrid 2833 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆))
120 dvn0 25867 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
12176, 119, 120syl2anc 583 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
122121adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
123116, 122eqtrd 2768 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑘 = 0) → (𝐶‘𝑘) = 𝐹)
124123fveq1d 6899 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥))
125124adantlr 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = (𝐹‘𝑥))
126 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
127117fvmpt2 7016 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝐹‘𝑥) = 𝐴)
128126, 77, 127syl2anc 583 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = 𝐴)
129128adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐹‘𝑥) = 𝐴)
130125, 129eqtrd 2768 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐶‘𝑘)‘𝑥) = 𝐴)
131 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
132 0m0e0 12363 . . . . . . . . . . . . . . . . . . . 20 (0 − 0) = 0
133132a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (0 − 0) = 0)
134131, 133eqtrd 2768 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (0 − 𝑘) = 0)
135134fveq2d 6901 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝐷‘(0 − 𝑘)) = (𝐷‘0))
136135fveq1d 6899 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
137136adantl 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
138137adantlr 714 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = ((𝐷‘0)‘𝑥))
139 dvnmul.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
140 fveq2 6897 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑛))
141140cbvmptv 5261 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
142139, 141eqtri 2756 . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))
143142fveq1i 6898 . . . . . . . . . . . . . . . . . 18 (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0)
144143a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐷‘0) = ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0))
145 eqidd 2729 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)) = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
146 fveq2 6897 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 0 → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
147146adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘0))
148 dvnmul.f . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)
14978, 84, 86, 74mptelpm 44549 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (ℂ ↑pm 𝑆))
150148, 149eqeltrid 2833 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 𝐺 ∈ (ℂ ↑pm 𝑆))
151 dvn0 25867 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
15276, 150, 151syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
153152adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘0) = 𝐺)
154147, 153eqtrd 2768 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑛 = 0) → ((𝑆 D𝑛 𝐺)‘𝑛) = 𝐺)
155148a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵))
156 mptexg 7233 . . . . . . . . . . . . . . . . . . . 20 (𝑋 ∈ 𝒫 𝑆 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V)
15782, 156syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ V)
158155, 157eqeltrd 2829 . . . . . . . . . . . . . . . . . 18 (𝜑 → 𝐺 ∈ V)
159145, 154, 112, 158fvmptd 7012 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛))‘0) = 𝐺)
160144, 159eqtrd 2768 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐷‘0) = 𝐺)
161160fveq1d 6899 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥))
162161ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘0)‘𝑥) = (𝐺‘𝑥))
163155, 78fvmpt2d 7018 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) = 𝐵)
164163adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (𝐺‘𝑥) = 𝐵)
165138, 162, 1643eqtrd 2772 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((𝐷‘(0 − 𝑘))‘𝑥) = 𝐵)
166130, 165oveq12d 7438 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)) = (𝐴 · 𝐵))
167103, 166oveq12d 7438 . . . . . . . . . . 11 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (1 · (𝐴 · 𝐵)))
16879mullidd 11263 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
169168adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → (1 · (𝐴 · 𝐵)) = (𝐴 · 𝐵))
170167, 169eqtrd 2768 . . . . . . . . . 10 (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 = 0) → ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
171 0re 11247 . . . . . . . . . . 11 0 ∈ ℝ
172171a1i 11 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
17395, 96, 170, 172, 79sumsnd 44388 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ {0} ((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))) = (𝐴 · 𝐵))
17494, 173eqtr2d 2769 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 · 𝐵) = Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))
175174mpteq2dva 5248 . . . . . . 7 (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
17689, 175eqtrd 2768 . . . . . 6 (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥)))))
177176a1i 11 . . . . 5 (𝑁 ∈ (ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘0) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...0)((0C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(0 − 𝑘))‘𝑥))))))
178 simp3 1136 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝜑)
179 simp1 1134 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → 𝑖 ∈ (0..^𝑁))
180 simp2 1135 . . . . . . . 8 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
181 pm3.35 802 . . . . . . . 8 ((𝜑 ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
182178, 180, 181syl2anc 583 . . . . . . 7 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
18376adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ⊆ ℂ)
18487adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆))
185 elfzonn0 13710 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
186185adantl 481 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
187 dvnp1 25868 . . . . . . . . . 10 ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑖 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
188183, 184, 186, 187syl3anc 1369 . . . . . . . . 9 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
189188adantr 480 . . . . . . . 8 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)))
190 simpr 484 . . . . . . . . 9 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))))
191190oveq2d 7436 . . . . . . . 8 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))))
192 eqid 2728 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
193 eqid 2728 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
19474adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑆 ∈ {ℝ, ℂ})
19581adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
196 fzfid 13971 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (0...𝑖) ∈ Fin)
197185adantr 480 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℕ0)
198 elfzelz 13534 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℤ)
199198adantl 481 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℤ)
200197, 199bccld 44697 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
201200nn0cnd 12565 . . . . . . . . . . . . . 14 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
202201adantll 713 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
2032023adant3 1130 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ)
204 simpll 766 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝜑)
205 0zd 12601 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℤ)
206 elfzoel2 13664 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
207206adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℤ)
208 elfzle1 13537 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑖) → 0 ≤ 𝑘)
209208adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ 𝑘)
210199zred 12697 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℝ)
211206zred 12697 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℝ)
212211adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑁 ∈ ℝ)
213185nn0red 12564 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
214213adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
215 elfzle2 13538 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ 𝑖)
216215adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑖)
217 elfzolt2 13674 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 < 𝑁)
218217adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 < 𝑁)
219210, 214, 212, 216, 218lelttrd 11403 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 < 𝑁)
220210, 212, 219ltled 11393 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ 𝑁)
221205, 207, 199, 209, 220elfzd 13525 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
222221adantll 713 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ (0...𝑁))
223 dvnmul.dvnf . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)
224106a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘)))
225 fvexd 6912 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘) ∈ V)
226224, 225fvmpt2d 7018 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
227226feq1d 6707 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ))
228223, 227mpbird 257 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ)
229204, 222, 228syl2anc 583 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘):𝑋⟶ℂ)
2302293adant3 1130 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐶‘𝑘):𝑋⟶ℂ)
231 simp3 1136 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
232230, 231ffvelcdmd 7095 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
233185nn0zd 12615 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
234233adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℤ)
235234, 199zsubcld 12702 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℤ)
236 elfzel2 13532 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℤ)
237236zred 12697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑖 ∈ ℝ)
238198zred 12697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℝ)
239237, 238subge0d 11835 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → (0 ≤ (𝑖 − 𝑘) ↔ 𝑘 ≤ 𝑖))
240215, 239mpbird 257 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑖 − 𝑘))
241240adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑖 − 𝑘))
242214, 210resubcld 11673 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℝ)
243212, 210resubcld 11673 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ∈ ℝ)
244171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ∈ ℝ)
245212, 244jca 511 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 ∈ ℝ ∧ 0 ∈ ℝ))
246 resubcl 11555 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 − 0) ∈ ℝ)
247245, 246syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) ∈ ℝ)
248214, 212, 210, 218ltsub1dd 11857 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 𝑘))
249244, 210, 212, 209lesub2dd 11862 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ (𝑁 − 0))
250242, 243, 247, 248, 249ltletrd 11405 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < (𝑁 − 0))
251211recnd 11273 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
252251subid1d 11591 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑁 − 0) = 𝑁)
253252adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 0) = 𝑁)
254250, 253breqtrd 5174 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) < 𝑁)
255242, 212, 254ltled 11393 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ≤ 𝑁)
256205, 207, 235, 241, 255elfzd 13525 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁))
257256adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ (0...𝑁))
258 ovex 7453 . . . . . . . . . . . . . . . . . 18 (𝑖 − 𝑘) ∈ V
259 eleq1 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ (𝑖 − 𝑘) ∈ (0...𝑁)))
260259anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁))))
261 fveq2 6897 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
262261feq1d 6707 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ))
263260, 262imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)))
264 nfv 1910 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
265 eleq1 2817 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑘 ∈ (0...𝑁) ↔ 𝑗 ∈ (0...𝑁)))
266265anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑁))))
267 fveq2 6897 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝑆 D𝑛 𝐺)‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑗))
268267feq1d 6707 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ))
269266, 268imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)))
270 dvnmul.dvng . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)
271264, 269, 270chvarfv 2229 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ)
272258, 263, 271vtocl 3543 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖 − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)
273204, 257, 272syl2anc 583 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ)
274 fveq2 6897 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑖 − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
275 fvexd 6912 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) ∈ V)
276142, 274, 256, 275fvmptd3 7028 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
277276adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
278277feq1d 6707 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)):𝑋⟶ℂ))
279273, 278mpbird 257 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ)
2802793adant3 1130 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (𝐷‘(𝑖 − 𝑘)):𝑋⟶ℂ)
281280, 231ffvelcdmd 7095 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ)
282232, 281mulcld 11265 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
283203, 282mulcld 11265 . . . . . . . . . . 11 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ)
2842033expa 1116 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (𝑖C𝑘) ∈ ℂ)
285234peano2zd 12700 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℤ)
286285, 199zsubcld 12702 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
287 peano2re 11418 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ ℝ → (𝑖 + 1) ∈ ℝ)
288237, 287syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → (𝑖 + 1) ∈ ℝ)
289 peano2re 11418 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
290238, 289syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ∈ ℝ)
291238ltp1d 12175 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑘 + 1))
292 1red 11246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...𝑖) → 1 ∈ ℝ)
293238, 237, 292, 215leadd1dd 11859 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑖) → (𝑘 + 1) ≤ (𝑖 + 1))
294238, 290, 288, 291, 293ltletrd 11405 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑖) → 𝑘 < (𝑖 + 1))
295238, 288, 294ltled 11393 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝑖) → 𝑘 ≤ (𝑖 + 1))
296295adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ≤ (𝑖 + 1))
297214, 287syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℝ)
298297, 210subge0d 11835 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
299296, 298mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ ((𝑖 + 1) − 𝑘))
300297, 210resubcld 11673 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
301 elfzop1le2 13678 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ≤ 𝑁)
302301adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ≤ 𝑁)
303297, 212, 210, 302lesub1dd 11861 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘))
304249, 253breqtrd 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑁 − 𝑘) ≤ 𝑁)
305300, 243, 212, 303, 304letrd 11402 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
306205, 207, 286, 299, 305elfzd 13525 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
307306adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
308 ovex 7453 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) − 𝑘) ∈ V
309 eleq1 2817 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝑗 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)))
310309anbi2d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))))
311 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝑆 D𝑛 𝐺)‘𝑗) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
312311feq1d 6707 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
313310, 312imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
314308, 313, 271vtocl 3543 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
315204, 307, 314syl2anc 583 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
316142a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐷 = (𝑛 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑛)))
317 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → 𝑛 = ((𝑖 + 1) − 𝑘))
318317fveq2d 6901 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑛 = ((𝑖 + 1) − 𝑘)) → ((𝑆 D𝑛 𝐺)‘𝑛) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
319 fvexd 6912 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) ∈ V)
320316, 318, 307, 319fvmptd 7012 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
321320feq1d 6707 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
322315, 321mpbird 257 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
323322ffvelcdmda 7094 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
3242323expa 1116 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
325323, 324mulcomd 11266 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
326325oveq2d 7436 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) = ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
327199peano2zd 12700 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℤ)
328171a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 ∈ ℝ)
329328, 238, 290, 208, 291lelttrd 11403 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (0...𝑖) → 0 < (𝑘 + 1))
330328, 290, 329ltled 11393 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑖) → 0 ≤ (𝑘 + 1))
331330adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 0 ≤ (𝑘 + 1))
332210, 289syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ ℝ)
333293adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ (𝑖 + 1))
334332, 297, 212, 333, 302letrd 11402 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ≤ 𝑁)
335205, 207, 327, 331, 334elfzd 13525 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
336335adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 + 1) ∈ (0...𝑁))
337 ovex 7453 . . . . . . . . . . . . . . . . . . 19 (𝑘 + 1) ∈ V
338 eleq1 2817 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝑗 ∈ (0...𝑁) ↔ (𝑘 + 1) ∈ (0...𝑁)))
339338anbi2d 629 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁))))
340 fveq2 6897 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑘 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑘 + 1)))
341340feq1d 6707 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑘 + 1) → ((𝐶‘𝑗):𝑋⟶ℂ ↔ (𝐶‘(𝑘 + 1)):𝑋⟶ℂ))
342339, 341imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)))
343 nfv 1910 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (0...𝑁))
344 nfmpt1 5256 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))
345106, 344nfcxfr 2897 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘𝐶
346 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘𝑗
347345, 346nffv 6907 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘(𝐶‘𝑗)
348 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘𝑋
349 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘ℂ
350347, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘(𝐶‘𝑗):𝑋⟶ℂ
351343, 350nfim 1892 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)
352 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑗 → (𝐶‘𝑘) = (𝐶‘𝑗))
353352feq1d 6707 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘𝑗):𝑋⟶ℂ))
354266, 353imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)))
355351, 354, 228chvarfv 2229 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐶‘𝑗):𝑋⟶ℂ)
356337, 342, 355vtocl 3543 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
357204, 336, 356syl2anc 583 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)):𝑋⟶ℂ)
358357ffvelcdmda 7094 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑘 + 1))‘𝑥) ∈ ℂ)
3592813expa 1116 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 − 𝑘))‘𝑥) ∈ ℂ)
360358, 359mulcld 11265 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
361323, 324mulcld 11265 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)) ∈ ℂ)
362360, 361addcld 11264 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) ∈ ℂ)
363326, 362eqeltrrd 2830 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
364284, 363mulcld 11265 . . . . . . . . . . . 12 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
3653643impa 1108 . . . . . . . . . . 11 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) ∈ ℂ)
366204, 74syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ∈ {ℝ, ℂ})
367171a1i 11 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
368204, 81syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
369366, 368, 202dvmptconst 45303 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝑖C𝑘))) = (𝑥 ∈ 𝑋 ↦ 0))
3702823expa 1116 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
371204, 222, 226syl2anc 583 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘))
372371eqcomd 2734 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘𝑘) = (𝐶‘𝑘))
373229feqmptd 6967 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘𝑘) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)))
374372, 373eqtr2d 2769 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥)) = ((𝑆 D𝑛 𝐹)‘𝑘))
375374oveq2d 7436 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
376366, 75syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑆 ⊆ ℂ)
377204, 119syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
378 elfznn0 13627 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → 𝑘 ∈ ℕ0)
379378adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℕ0)
380 dvnp1 25868 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
381376, 377, 379, 380syl3anc 1369 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)))
382381eqcomd 2734 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑘)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
383 fveq2 6897 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑘 + 1) → ((𝑆 D𝑛 𝐹)‘𝑛) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
384 fvexd 6912 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) ∈ V)
385109, 383, 336, 384fvmptd3 7028 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)))
386385eqcomd 2734 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝐶‘(𝑘 + 1)))
387357feqmptd 6967 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐶‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
388386, 387eqtrd 2768 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐹)‘(𝑘 + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
389375, 382, 3883eqtrd 2772 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐶‘𝑘)‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐶‘(𝑘 + 1))‘𝑥)))
390277eqcomd 2734 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − 𝑘)))
391279feqmptd 6967 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘(𝑖 − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
392390, 391eqtr2d 2769 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘)))
393392oveq2d 7436 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
394204, 150syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → 𝐺 ∈ (ℂ ↑pm 𝑆))
395 fznn0sub 13566 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...𝑖) → (𝑖 − 𝑘) ∈ ℕ0)
396395adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 − 𝑘) ∈ ℕ0)
397 dvnp1 25868 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ ℂ ∧ 𝐺 ∈ (ℂ ↑pm 𝑆) ∧ (𝑖 − 𝑘) ∈ ℕ0) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
398376, 394, 396, 397syl3anc 1369 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))))
399398eqcomd 2734 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D ((𝑆 D𝑛 𝐺)‘(𝑖 − 𝑘))) = ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)))
400214recnd 11273 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑖 ∈ ℂ)
401 1cnd 11240 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 1 ∈ ℂ)
402210recnd 11273 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → 𝑘 ∈ ℂ)
403400, 401, 402addsubd 11623 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1) − 𝑘) = ((𝑖 − 𝑘) + 1))
404403eqcomd 2734 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 − 𝑘) + 1) = ((𝑖 + 1) − 𝑘))
405404fveq2d 6901 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
406405adantll 713 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)))
407320eqcomd 2734 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 𝑘)))
408322feqmptd 6967 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
409406, 407, 4083eqtrd 2772 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑆 D𝑛 𝐺)‘((𝑖 − 𝑘) + 1)) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
410393, 399, 4093eqtrd 2772 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
411366, 324, 358, 389, 359, 323, 410dvmptmul 25906 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥)))))
412366, 284, 367, 369, 370, 362, 411dvmptmul 25906 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)))))
413370mul02d 11443 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = 0)
414326oveq1d 7435 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)))
415363, 284mulcomd 11266 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
416414, 415eqtrd 2768 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
417413, 416oveq12d 7438 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
418364addlidd 11446 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → (0 + ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
419417, 418eqtrd 2768 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘))) = ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
420419mpteq2dva 5248 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑥 ∈ 𝑋 ↦ ((0 · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + (((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) · ((𝐶‘𝑘)‘𝑥))) · (𝑖C𝑘)))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
421412, 420eqtrd 2768 . . . . . . . . . . 11 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
422192, 193, 194, 195, 196, 283, 365, 421dvmptfsum 25920 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
423202adantlr 714 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖C𝑘) ∈ ℂ)
424360an32s 651 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) ∈ ℂ)
425 anass 468 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)))
426 ancom 460 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖)))
427426anbi2i 622 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))))
428 anass 468 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))))
429428bicomi 223 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝑖))) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
430427, 429bitri 275 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑥 ∈ 𝑋)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
431425, 430bitri 275 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)))
432431imbi1i 349 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ))
433324, 432mpbi 229 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
434431imbi1i 349 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ))
435323, 434mpbi 229 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
436433, 435mulcld 11265 . . . . . . . . . . . . . 14 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
437423, 424, 436adddid 11269 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
438437sumeq2dv 15682 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
439196adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...𝑖) ∈ Fin)
440423, 424mulcld 11265 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ)
441423, 436mulcld 11265 . . . . . . . . . . . . 13 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
442439, 440, 441fsumadd 15719 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
443 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑘 = ℎ → (𝑖C𝑘) = (𝑖Cℎ))
444 fvoveq1 7443 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = ℎ → (𝐶‘(𝑘 + 1)) = (𝐶‘(ℎ + 1)))
445444fveq1d 6899 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = ℎ → ((𝐶‘(𝑘 + 1))‘𝑥) = ((𝐶‘(ℎ + 1))‘𝑥))
446 oveq2 7428 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = ℎ → (𝑖 − 𝑘) = (𝑖 − ℎ))
447446fveq2d 6901 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = ℎ → (𝐷‘(𝑖 − 𝑘)) = (𝐷‘(𝑖 − ℎ)))
448447fveq1d 6899 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = ℎ → ((𝐷‘(𝑖 − 𝑘))‘𝑥) = ((𝐷‘(𝑖 − ℎ))‘𝑥))
449445, 448oveq12d 7438 . . . . . . . . . . . . . . . . . . 19 (𝑘 = ℎ → (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) = (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
450443, 449oveq12d 7438 . . . . . . . . . . . . . . . . . 18 (𝑘 = ℎ → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))))
451 nfcv 2899 . . . . . . . . . . . . . . . . . 18 Ⅎℎ(0...𝑖)
452 nfcv 2899 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘(0...𝑖)
453 nfcv 2899 . . . . . . . . . . . . . . . . . 18 Ⅎℎ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)))
454 nfcv 2899 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(𝑖Cℎ)
455 nfcv 2899 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘 ·
456 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘(ℎ + 1)
457345, 456nffv 6907 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘(𝐶‘(ℎ + 1))
458 nfcv 2899 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘𝑥
459457, 458nffv 6907 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝐶‘(ℎ + 1))‘𝑥)
460 nfmpt1 5256 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))
461139, 460nfcxfr 2897 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘𝐷
462 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘(𝑖 − ℎ)
463461, 462nffv 6907 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘(𝐷‘(𝑖 − ℎ))
464463, 458nffv 6907 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝐷‘(𝑖 − ℎ))‘𝑥)
465459, 455, 464nfov 7450 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))
466454, 455, 465nfov 7450 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
467450, 451, 452, 453, 466cbvsum 15674 . . . . . . . . . . . . . . . . 17 Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)))
468467a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))))
469 1zzd 12624 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℤ)
47090a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℤ)
471233ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ ℤ)
472 nfv 1910 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)
473 nfcv 2899 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘ℎ
474473, 452nfel 2914 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘 ℎ ∈ (0...𝑖)
475472, 474nfan 1895 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖))
476466, 349nfel 2914 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ
477475, 476nfim 1892 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)
478 eleq1 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = ℎ → (𝑘 ∈ (0...𝑖) ↔ ℎ ∈ (0...𝑖)))
479478anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑘 = ℎ → ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) ↔ (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖))))
480450eleq1d 2814 . . . . . . . . . . . . . . . . . . 19 (𝑘 = ℎ → (((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ ↔ ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ))
481479, 480imbi12d 344 . . . . . . . . . . . . . . . . . 18 (𝑘 = ℎ → (((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)))
482477, 481, 440chvarfv 2229 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ ℎ ∈ (0...𝑖)) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) ∈ ℂ)
483 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (ℎ = (𝑗 − 1) → (𝑖Cℎ) = (𝑖C(𝑗 − 1)))
484 fvoveq1 7443 . . . . . . . . . . . . . . . . . . . 20 (ℎ = (𝑗 − 1) → (𝐶‘(ℎ + 1)) = (𝐶‘((𝑗 − 1) + 1)))
485484fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (ℎ = (𝑗 − 1) → ((𝐶‘(ℎ + 1))‘𝑥) = ((𝐶‘((𝑗 − 1) + 1))‘𝑥))
486 oveq2 7428 . . . . . . . . . . . . . . . . . . . . 21 (ℎ = (𝑗 − 1) → (𝑖 − ℎ) = (𝑖 − (𝑗 − 1)))
487486fveq2d 6901 . . . . . . . . . . . . . . . . . . . 20 (ℎ = (𝑗 − 1) → (𝐷‘(𝑖 − ℎ)) = (𝐷‘(𝑖 − (𝑗 − 1))))
488487fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (ℎ = (𝑗 − 1) → ((𝐷‘(𝑖 − ℎ))‘𝑥) = ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))
489485, 488oveq12d 7438 . . . . . . . . . . . . . . . . . 18 (ℎ = (𝑗 − 1) → (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥)) = (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
490483, 489oveq12d 7438 . . . . . . . . . . . . . . . . 17 (ℎ = (𝑗 − 1) → ((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
491469, 470, 471, 482, 490fsumshft 15759 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σℎ ∈ (0...𝑖)((𝑖Cℎ) · (((𝐶‘(ℎ + 1))‘𝑥) · ((𝐷‘(𝑖 − ℎ))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
492468, 491eqtrd 2768 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
493 0p1e1 12365 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
494493oveq1i 7430 . . . . . . . . . . . . . . . . 17 ((0 + 1)...(𝑖 + 1)) = (1...(𝑖 + 1))
495494sumeq1i 15677 . . . . . . . . . . . . . . . 16 Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)))
496495a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((0 + 1)...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))))
497 elfzelz 13534 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℤ)
498497zcnd 12698 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℂ)
499 1cnd 11240 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℂ)
500498, 499npcand 11606 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
501500fveq2d 6901 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (1...(𝑖 + 1)) → (𝐶‘((𝑗 − 1) + 1)) = (𝐶‘𝑗))
502501fveq1d 6899 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (1...(𝑖 + 1)) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥))
503502adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘((𝑗 − 1) + 1))‘𝑥) = ((𝐶‘𝑗)‘𝑥))
504213recnd 11273 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℂ)
505504adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℂ)
506498adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℂ)
507499adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℂ)
508505, 506, 507subsub3d 11632 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 − (𝑗 − 1)) = ((𝑖 + 1) − 𝑗))
509508fveq2d 6901 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘(𝑖 − (𝑗 − 1))) = (𝐷‘((𝑖 + 1) − 𝑗)))
510509fveq1d 6899 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
511503, 510oveq12d 7438 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥)) = (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
512511oveq2d 7436 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
513512sumeq2dv 15682 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
514513ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
515 nfv 1910 . . . . . . . . . . . . . . . . 17 Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋)
516 nfcv 2899 . . . . . . . . . . . . . . . . 17 Ⅎ𝑗((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
517 fzfid 13971 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...(𝑖 + 1)) ∈ Fin)
518185adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℕ0)
519497adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℤ)
520 1zzd 12624 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℤ)
521519, 520zsubcld 12702 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑗 − 1) ∈ ℤ)
522518, 521bccld 44697 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
523522nn0cnd 12565 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
524523adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
525524adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖C(𝑗 − 1)) ∈ ℂ)
5261ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝜑)
527 0zd 12601 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ∈ ℤ)
528206adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℤ)
529171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ∈ ℝ)
530497zred 12697 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ)
531 1red 11246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ∈ ℝ)
532 0lt1 11767 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 < 1
533532a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 1)
534 elfzle1 13537 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...(𝑖 + 1)) → 1 ≤ 𝑗)
535529, 531, 530, 533, 534ltletrd 11405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 0 < 𝑗)
536529, 530, 535ltled 11393 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...(𝑖 + 1)) → 0 ≤ 𝑗)
537536adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ 𝑗)
538530adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ)
539213adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℝ)
540 1red 11246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 1 ∈ ℝ)
541539, 540readdcld 11274 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
542211adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑁 ∈ ℝ)
543 elfzle2 13538 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ≤ (𝑖 + 1))
544543adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ (𝑖 + 1))
545301adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
546538, 541, 542, 544, 545letrd 11402 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ≤ 𝑁)
547527, 528, 519, 537, 546elfzd 13525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
548547adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ (0...𝑁))
549526, 548, 355syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ)
550549adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐶‘𝑗):𝑋⟶ℂ)
551 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑥 ∈ 𝑋)
552550, 551ffvelcdmd 7095 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ)
553233adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑖 ∈ ℤ)
554553peano2zd 12700 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
555554, 519zsubcld 12702 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℤ)
556541, 538subge0d 11835 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑗) ↔ 𝑗 ≤ (𝑖 + 1)))
557544, 556mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑗))
558541, 538resubcld 11673 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ ℝ)
559558leidd 11811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ ((𝑖 + 1) − 𝑗))
560530, 535elrpd 13046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (1...(𝑖 + 1)) → 𝑗 ∈ ℝ+)
561560adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → 𝑗 ∈ ℝ+)
562541, 561ltsubrpd 13081 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < (𝑖 + 1))
563558, 541, 542, 562, 545ltletrd 11405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
564558, 558, 542, 559, 563lelttrd 11403 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) < 𝑁)
565558, 542, 564ltled 11393 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ≤ 𝑁)
566527, 528, 555, 557, 565elfzd 13525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
567566adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
568 nfv 1910 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))
569 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Ⅎ𝑘((𝑖 + 1) − 𝑗)
570461, 569nffv 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗))
571570, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ
572568, 571nfim 1892 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
573 ovex 7453 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 + 1) − 𝑗) ∈ V
574 eleq1 2817 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝑘 ∈ (0...𝑁) ↔ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)))
575574anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁))))
576 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = ((𝑖 + 1) − 𝑗) → (𝐷‘𝑘) = (𝐷‘((𝑖 + 1) − 𝑗)))
577576feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = ((𝑖 + 1) − 𝑗) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ))
578575, 577imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = ((𝑖 + 1) − 𝑗) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)))
579139a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘)))
580 fvexd 6912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘) ∈ V)
581579, 580fvmpt2d 7018 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘) = ((𝑆 D𝑛 𝐺)‘𝑘))
582581feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ))
583270, 582mpbird 257 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ)
584572, 573, 578, 583vtoclf 3549 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑖 + 1) − 𝑗) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
585526, 567, 584syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
586585adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑗)):𝑋⟶ℂ)
587586, 551ffvelcdmd 7095 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
588552, 587mulcld 11265 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
589525, 588mulcld 11265 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...(𝑖 + 1))) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
590 1zzd 12624 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℤ)
591233peano2zd 12700 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℤ)
592493eqcomi 2737 . . . . . . . . . . . . . . . . . . . . . . 23 1 = (0 + 1)
593592a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → 1 = (0 + 1))
594171a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ∈ ℝ)
595 1red 11246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℝ)
596185nn0ge0d 12566 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 0 ≤ 𝑖)
597594, 213, 595, 596leadd1dd 11859 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (0 + 1) ≤ (𝑖 + 1))
598593, 597eqbrtrd 5170 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 1 ≤ (𝑖 + 1))
599590, 591, 5983jca 1126 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
600 eluz2 12859 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ (ℤ≥‘1) ↔ (1 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 1 ≤ (𝑖 + 1)))
601599, 600sylibr 233 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ≥‘1))
602 eluzfz2 13542 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ≥‘1) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
603601, 602syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
604603ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (1...(𝑖 + 1)))
605 oveq1 7427 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → (𝑗 − 1) = ((𝑖 + 1) − 1))
606605oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (𝑖C(𝑗 − 1)) = (𝑖C((𝑖 + 1) − 1)))
607 fveq2 6897 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐶‘𝑗) = (𝐶‘(𝑖 + 1)))
608607fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
609 oveq2 7428 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑖 + 1) → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − (𝑖 + 1)))
610609fveq2d 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
611610fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
612608, 611oveq12d 7438 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑖 + 1) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
613606, 612oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑖 + 1) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
614515, 516, 517, 589, 604, 613fsumsplit1 15724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
615 1cnd 11240 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → 1 ∈ ℂ)
616504, 615pncand 11603 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 1) = 𝑖)
617616oveq2d 7436 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = (𝑖C𝑖))
618 bcnn 14304 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖C𝑖) = 1)
619185, 618syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑖C𝑖) = 1)
620617, 619eqtrd 2768 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑖C((𝑖 + 1) − 1)) = 1)
621504, 615addcld 11264 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℂ)
622621subidd 11590 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − (𝑖 + 1)) = 0)
623622fveq2d 6901 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
624623fveq1d 6899 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) = ((𝐷‘0)‘𝑥))
625624oveq2d 7436 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
626620, 625oveq12d 7438 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))))
627626ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))))
628 simpl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝜑)
629 fzofzp1 13762 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
630629adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
631 nfv 1910 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))
632 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Ⅎ𝑘(𝑖 + 1)
633345, 632nffv 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝐶‘(𝑖 + 1))
634633, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝐶‘(𝑖 + 1)):𝑋⟶ℂ
635631, 634nfim 1892 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
636 ovex 7453 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 + 1) ∈ V
637 eleq1 2817 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑖 + 1) ∈ (0...𝑁)))
638637anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁))))
639 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = (𝑖 + 1) → (𝐶‘𝑘) = (𝐶‘(𝑖 + 1)))
640639feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘(𝑖 + 1)):𝑋⟶ℂ))
641638, 640imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)))
642635, 636, 641, 228vtoclf 3549 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
643628, 630, 642syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘(𝑖 + 1)):𝑋⟶ℂ)
644643ffvelcdmda 7094 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑖 + 1))‘𝑥) ∈ ℂ)
645 nfv 1910 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝜑 ∧ 0 ∈ (0...𝑁))
646 nfcv 2899 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Ⅎ𝑘0
647461, 646nffv 6907 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Ⅎ𝑘(𝐷‘0)
648647, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝐷‘0):𝑋⟶ℂ
649645, 648nfim 1892 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
650 c0ex 11239 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
651 eleq1 2817 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝑘 ∈ (0...𝑁) ↔ 0 ∈ (0...𝑁)))
652651anbi2d 629 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 0 ∈ (0...𝑁))))
653 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 = 0 → (𝐷‘𝑘) = (𝐷‘0))
654653feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
655652, 654imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)))
656649, 650, 655, 583vtoclf 3549 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐷‘0):𝑋⟶ℂ)
6571, 112, 656syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐷‘0):𝑋⟶ℂ)
658657adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘0):𝑋⟶ℂ)
659658ffvelcdmda 7094 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘0)‘𝑥) ∈ ℂ)
660644, 659mulcld 11265 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) ∈ ℂ)
661660mullidd 11263 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
662627, 661eqtrd 2768 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
663 1m1e0 12315 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 − 1) = 0
664663fveq2i 6900 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
6653eqcomi 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℤ≥‘0) = ℕ0
666664, 665eqtr2i 2757 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ0 = (ℤ≥‘(1 − 1))
667666a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → ℕ0 = (ℤ≥‘(1 − 1)))
668185, 667eleqtrd 2831 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘(1 − 1)))
669 fzdifsuc2 44692 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (ℤ≥‘(1 − 1)) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
670668, 669syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (1...𝑖) = ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
671670eqcomd 2734 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)}) = (1...𝑖))
672671sumeq1d 15680 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
673672ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))))
674662, 673oveq12d 7438 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C((𝑖 + 1) − 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑗 ∈ ((1...(𝑖 + 1)) ∖ {(𝑖 + 1)})((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
675514, 614, 6743eqtrd 2772 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...(𝑖 + 1))((𝑖C(𝑗 − 1)) · (((𝐶‘((𝑗 − 1) + 1))‘𝑥) · ((𝐷‘(𝑖 − (𝑗 − 1)))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
676492, 496, 6753eqtrd 2772 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))))
677 nfcv 2899 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘(𝑖C0)
678345, 646nffv 6907 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(𝐶‘0)
679678, 458nffv 6907 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘((𝐶‘0)‘𝑥)
680 nfcv 2899 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝑖 + 1) − 0)
681461, 680nffv 6907 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(𝐷‘((𝑖 + 1) − 0))
682681, 458nffv 6907 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘((𝐷‘((𝑖 + 1) − 0))‘𝑥)
683679, 455, 682nfov 7450 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘(((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
684677, 455, 683nfov 7450 . . . . . . . . . . . . . . . 16 Ⅎ𝑘((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
685665a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (ℤ≥‘0) = ℕ0)
686185, 685eleqtrrd 2832 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (ℤ≥‘0))
687 eluzfz1 13541 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑖))
688686, 687syl 17 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → 0 ∈ (0...𝑖))
689688ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑖))
690 oveq2 7428 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (𝑖C𝑘) = (𝑖C0))
691104fveq1d 6899 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘0)‘𝑥))
692 oveq2 7428 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − 0))
693692fveq2d 6901 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − 0)))
694693fveq1d 6899 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − 0))‘𝑥))
695691, 694oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
696690, 695oveq12d 7438 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
697472, 684, 439, 441, 689, 696fsumsplit1 15724 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
698621subid1d 11591 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1) − 0) = (𝑖 + 1))
699698fveq2d 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝐷‘((𝑖 + 1) − 0)) = (𝐷‘(𝑖 + 1)))
700699fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝐷‘((𝑖 + 1) − 0))‘𝑥) = ((𝐷‘(𝑖 + 1))‘𝑥))
701700oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
702701oveq2d 7436 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
703702oveq1d 7435 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (0..^𝑁) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
704703ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
705 bcn0 14302 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0 → (𝑖C0) = 1)
706185, 705syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖C0) = 1)
707706oveq1d 7435 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
708707ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
709678, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝐶‘0):𝑋⟶ℂ
710645, 709nfim 1892 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
711104feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → ((𝐶‘𝑘):𝑋⟶ℂ ↔ (𝐶‘0):𝑋⟶ℂ))
712652, 711imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐶‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)))
713710, 650, 712, 228vtoclf 3549 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 0 ∈ (0...𝑁)) → (𝐶‘0):𝑋⟶ℂ)
7141, 112, 713syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐶‘0):𝑋⟶ℂ)
715714adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐶‘0):𝑋⟶ℂ)
716715ffvelcdmda 7094 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘0)‘𝑥) ∈ ℂ)
717461, 632nffv 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝐷‘(𝑖 + 1))
718717, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(𝐷‘(𝑖 + 1)):𝑋⟶ℂ
719631, 718nfim 1892 . . . . . . . . . . . . . . . . . . . . . 22 Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
720 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = (𝑖 + 1) → (𝐷‘𝑘) = (𝐷‘(𝑖 + 1)))
721720feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑖 + 1) → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘(𝑖 + 1)):𝑋⟶ℂ))
722638, 721imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)))
723719, 636, 722, 583vtoclf 3549 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑖 + 1) ∈ (0...𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
724628, 630, 723syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘(𝑖 + 1)):𝑋⟶ℂ)
725724ffvelcdmda 7094 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘(𝑖 + 1))‘𝑥) ∈ ℂ)
726716, 725mulcld 11265 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) ∈ ℂ)
727726mullidd 11263 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
728708, 727eqtrd 2768 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) = (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)))
729 nfv 1910 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑗 𝑖 ∈ (0..^𝑁)
730 1zzd 12624 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ∈ ℤ)
731233adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑖 ∈ ℤ)
732 eldifi 4125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (0...𝑖))
733 elfzelz 13534 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
734732, 733syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℤ)
735734adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ ℤ)
736 elfznn0 13627 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℕ0)
737732, 736syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ0)
738 eldifsni 4794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≠ 0)
739737, 738jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((0...𝑖) ∖ {0}) → (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0))
740 elnnne0 12517 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0))
741739, 740sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ ℕ)
742 nnge1 12271 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ℕ → 1 ≤ 𝑗)
743741, 742syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 1 ≤ 𝑗)
744743adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 1 ≤ 𝑗)
745 elfzle2 13538 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
746732, 745syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ≤ 𝑖)
747746adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ≤ 𝑖)
748730, 731, 735, 744, 747elfzd 13525 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ ((0...𝑖) ∖ {0})) → 𝑗 ∈ (1...𝑖))
749748ex 412 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) → 𝑗 ∈ (1...𝑖)))
750 0zd 12601 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 0 ∈ ℤ)
751 elfzel2 13532 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑖 ∈ ℤ)
752 elfzelz 13534 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℤ)
753171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 ∈ ℝ)
754752zred 12697 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ℝ)
755 1red 11246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 1 ∈ ℝ)
756532a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 0 < 1)
757 elfzle1 13537 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (1...𝑖) → 1 ≤ 𝑗)
758753, 755, 754, 756, 757ltletrd 11405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (1...𝑖) → 0 < 𝑗)
759753, 754, 758ltled 11393 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 0 ≤ 𝑗)
760 elfzle2 13538 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≤ 𝑖)
761750, 751, 752, 759, 760elfzd 13525 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (0...𝑖))
762753, 758gtned 11380 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (1...𝑖) → 𝑗 ≠ 0)
763 nelsn 4669 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ≠ 0 → ¬ 𝑗 ∈ {0})
764762, 763syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (1...𝑖) → ¬ 𝑗 ∈ {0})
765761, 764eldifd 3958 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
766765adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ((0...𝑖) ∖ {0}))
767766ex 412 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ (1...𝑖) → 𝑗 ∈ ((0...𝑖) ∖ {0})))
768749, 767impbid 211 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
769729, 768alrimi 2202 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
770 dfcleq 2721 . . . . . . . . . . . . . . . . . . 19 (((0...𝑖) ∖ {0}) = (1...𝑖) ↔ ∀𝑗(𝑗 ∈ ((0...𝑖) ∖ {0}) ↔ 𝑗 ∈ (1...𝑖)))
771769, 770sylibr 233 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → ((0...𝑖) ∖ {0}) = (1...𝑖))
772771sumeq1d 15680 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
773772ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
774728, 773oveq12d 7438 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
775697, 704, 7743eqtrd 2772 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
776676, 775oveq12d 7438 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
777 fzfid 13971 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) ∈ Fin)
778185adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
779766, 734syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ ℤ)
780 1zzd 12624 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → 1 ∈ ℤ)
781779, 780zsubcld 12702 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑗 − 1) ∈ ℤ)
782778, 781bccld 44697 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℕ0)
783782nn0cnd 12565 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
784783adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
785784adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (𝑖C(𝑗 − 1)) ∈ ℂ)
786 simpl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋))
787 fzelp1 13586 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1...𝑖) → 𝑗 ∈ (1...(𝑖 + 1)))
788787adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → 𝑗 ∈ (1...(𝑖 + 1)))
789786, 788, 552syl2anc 583 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐶‘𝑗)‘𝑥) ∈ ℂ)
790788, 587syldan 590 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) ∈ ℂ)
791789, 790mulcld 11265 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) ∈ ℂ)
792785, 791mulcld 11265 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ (1...𝑖)) → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
793777, 792fsumcl 15712 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) ∈ ℂ)
794185adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑖 ∈ ℕ0)
795 elfzelz 13534 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ ℤ)
796795adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ ℤ)
797794, 796bccld 44697 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℕ0)
798797nn0cnd 12565 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
799798adantll 713 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
800799adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C𝑘) ∈ ℂ)
801 simpll 766 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝜑 ∧ 𝑖 ∈ (0..^𝑁)))
802 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑥 ∈ 𝑋)
803761ssriv 3984 . . . . . . . . . . . . . . . . . . . 20 (1...𝑖) ⊆ (0...𝑖)
804 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (1...𝑖))
805803, 804sselid 3978 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝑖) → 𝑘 ∈ (0...𝑖))
806805adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → 𝑘 ∈ (0...𝑖))
807801, 802, 806, 433syl21anc 837 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
808806, 435syldan 590 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
809807, 808mulcld 11265 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
810800, 809mulcld 11265 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
811777, 810fsumcl 15712 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
812660, 793, 726, 811add4d 11473 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
813 oveq1 7427 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1))
814813oveq2d 7436 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (𝑖C(𝑗 − 1)) = (𝑖C(𝑘 − 1)))
815 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐶‘𝑗) = (𝐶‘𝑘))
816815fveq1d 6899 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐶‘𝑗)‘𝑥) = ((𝐶‘𝑘)‘𝑥))
817 oveq2 7428 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑘 → ((𝑖 + 1) − 𝑗) = ((𝑖 + 1) − 𝑘))
818817fveq2d 6901 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑘 → (𝐷‘((𝑖 + 1) − 𝑗)) = (𝐷‘((𝑖 + 1) − 𝑘)))
819818fveq1d 6899 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑘 → ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥) = ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))
820816, 819oveq12d 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑘 → (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)) = (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
821814, 820oveq12d 7438 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → ((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
822 nfcv 2899 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(1...𝑖)
823 nfcv 2899 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑗(1...𝑖)
824 nfcv 2899 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘(𝑖C(𝑗 − 1))
825347, 458nffv 6907 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘((𝐶‘𝑗)‘𝑥)
826570, 458nffv 6907 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)
827825, 455, 826nfov 7450 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘(((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))
828824, 455, 827nfov 7450 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥)))
829 nfcv 2899 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑗((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
830821, 822, 823, 828, 829cbvsum 15674 . . . . . . . . . . . . . . . . . 18 Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))
831830a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) = Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
832831oveq1d 7435 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
833 peano2zm 12636 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
834796, 833syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑘 − 1) ∈ ℤ)
835794, 834bccld 44697 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℕ0)
836835nn0cnd 12565 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
837836adantll 713 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
838837adantlr 714 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖C(𝑘 − 1)) ∈ ℂ)
839838, 809mulcld 11265 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
840777, 839, 810fsumadd 15719 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
841840eqcomd 2734 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (1...𝑖)((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
842836, 798addcomd 11447 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) = ((𝑖C𝑘) + (𝑖C(𝑘 − 1))))
843 bcpasc 14313 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
844794, 796, 843syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖C𝑘) + (𝑖C(𝑘 − 1))) = ((𝑖 + 1)C𝑘))
845842, 844eqtr2d 2769 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) = ((𝑖C(𝑘 − 1)) + (𝑖C𝑘)))
846845oveq1d 7435 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
847846adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
848847adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
849838, 800, 809adddird 11270 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) + (𝑖C𝑘)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
850848, 849eqtr2d 2769 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
851850sumeq2dv 15682 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖C(𝑘 − 1)) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) + ((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
852832, 841, 8513eqtrd 2772 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
853852oveq2d 7436 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
854 peano2nn0 12543 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0)
855794, 854syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → (𝑖 + 1) ∈ ℕ0)
856855, 796bccld 44697 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
857856nn0cnd 12565 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
858857adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
859858adantlr 714 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
860859, 809mulcld 11265 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (1...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
861777, 860fsumcl 15712 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
862660, 726, 861addassd 11267 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))))
863185, 854syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ ℕ0)
864 bcn0 14302 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C0) = 1)
865863, 864syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C0) = 1)
866865, 701oveq12d 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^𝑁) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
867866ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) = (1 · (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))))
868867, 727eqtr2d 2769 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
869771ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((0...𝑖) ∖ {0}) = (1...𝑖))
870869eqcomd 2734 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1...𝑖) = ((0...𝑖) ∖ {0}))
871870sumeq1d 15680 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
872868, 871oveq12d 7438 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
873 nfcv 2899 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘((𝑖 + 1)C0)
874873, 455, 683nfov 7450 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘(((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥)))
875197, 854syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → (𝑖 + 1) ∈ ℕ0)
876875, 199bccld 44697 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
877876nn0cnd 12565 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
878877adantll 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
879878adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → ((𝑖 + 1)C𝑘) ∈ ℂ)
880879, 436mulcld 11265 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝑖)) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
881 oveq2 7428 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C0))
882881, 695oveq12d 7438 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))))
883472, 874, 439, 880, 689, 882fsumsplit1 15724 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
884883eqcomd 2734 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C0) · (((𝐶‘0)‘𝑥) · ((𝐷‘((𝑖 + 1) − 0))‘𝑥))) + Σ𝑘 ∈ ((0...𝑖) ∖ {0})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
885872, 884eqtrd 2768 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
886885oveq2d 7436 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
887 bcnn 14304 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 + 1) ∈ ℕ0 → ((𝑖 + 1)C(𝑖 + 1)) = 1)
888863, 887syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
889888ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝑖 + 1)C(𝑖 + 1)) = 1)
890889oveq1d 7435 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
891623adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))) = (𝐷‘0))
892891feq1d 6707 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ ↔ (𝐷‘0):𝑋⟶ℂ))
893658, 892mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
894893adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝐷‘((𝑖 + 1) − (𝑖 + 1))):𝑋⟶ℂ)
895 simpr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
896894, 895ffvelcdmd 7095 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥) ∈ ℂ)
897644, 896mulcld 11265 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) ∈ ℂ)
898897mullidd 11263 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (1 · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
899625ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)))
900890, 898, 8993eqtrrd 2773 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
901 fzdifsuc 13594 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ≥‘0) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
902686, 901syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (0...𝑖) = ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)}))
903902sumeq1d 15680 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑁) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
904903ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
905900, 904oveq12d 7438 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + Σ𝑘 ∈ (0...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
906 nfcv 2899 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘((𝑖 + 1)C(𝑖 + 1))
907633, 458nffv 6907 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘((𝐶‘(𝑖 + 1))‘𝑥)
908 nfcv 2899 . . . . . . . . . . . . . . . . . . . . 21 Ⅎ𝑘((𝑖 + 1) − (𝑖 + 1))
909461, 908nffv 6907 . . . . . . . . . . . . . . . . . . . 20 Ⅎ𝑘(𝐷‘((𝑖 + 1) − (𝑖 + 1)))
910909, 458nffv 6907 . . . . . . . . . . . . . . . . . . 19 Ⅎ𝑘((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)
911907, 455, 910nfov 7450 . . . . . . . . . . . . . . . . . 18 Ⅎ𝑘(((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
912906, 455, 911nfov 7450 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘(((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
913 fzfid 13971 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (0...(𝑖 + 1)) ∈ Fin)
914863adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℕ0)
915 elfzelz 13534 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ∈ ℤ)
916915adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℤ)
917914, 916bccld 44697 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℕ0)
918917nn0cnd 12565 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
919918adantll 713 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
920919adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1)C𝑘) ∈ ℂ)
921628adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
92290a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℤ)
923206adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℤ)
924 elfzle1 13537 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...(𝑖 + 1)) → 0 ≤ 𝑘)
925924adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ 𝑘)
926916zred 12697 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ ℝ)
927914nn0red 12564 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℝ)
928211adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑁 ∈ ℝ)
929 elfzle2 13538 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ (0...(𝑖 + 1)) → 𝑘 ≤ (𝑖 + 1))
930929adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ (𝑖 + 1))
931301adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ≤ 𝑁)
932926, 927, 928, 930, 931letrd 11402 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ≤ 𝑁)
933922, 923, 916, 925, 932elfzd 13525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
934933adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑘 ∈ (0...𝑁))
935921, 934, 228syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ)
936935adantlr 714 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐶‘𝑘):𝑋⟶ℂ)
937 simplr 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝑥 ∈ 𝑋)
938936, 937ffvelcdmd 7095 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐶‘𝑘)‘𝑥) ∈ ℂ)
939921adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 𝜑)
940591adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑖 + 1) ∈ ℤ)
941940, 916zsubcld 12702 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℤ)
942927, 926subge0d 11835 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (0 ≤ ((𝑖 + 1) − 𝑘) ↔ 𝑘 ≤ (𝑖 + 1)))
943930, 942mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ≤ ((𝑖 + 1) − 𝑘))
944927, 926resubcld 11673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ ℝ)
945928, 926resubcld 11673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ∈ ℝ)
946928, 171, 246sylancl 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) ∈ ℝ)
947927, 928, 926, 931lesub1dd 11861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 𝑘))
948171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → 0 ∈ ℝ)
949948, 926, 928, 925lesub2dd 11862 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 𝑘) ≤ (𝑁 − 0))
950944, 945, 946, 947, 949letrd 11402 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ (𝑁 − 0))
951252adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝑁 − 0) = 𝑁)
952950, 951breqtrd 5174 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ≤ 𝑁)
953922, 923, 941, 943, 952elfzd 13525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
954953adantll 713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
955954adantlr 714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝑖 + 1) − 𝑘) ∈ (0...𝑁))
956 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = ((𝑖 + 1) − 𝑘) → (𝐷‘𝑗) = (𝐷‘((𝑖 + 1) − 𝑘)))
957956feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = ((𝑖 + 1) − 𝑘) → ((𝐷‘𝑗):𝑋⟶ℂ ↔ (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ))
958310, 957imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = ((𝑖 + 1) − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)))
959461, 346nffv 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝐷‘𝑗)
960959, 348, 349nff 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝐷‘𝑗):𝑋⟶ℂ
961343, 960nfim 1892 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)
962 fveq2 6897 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑗 → (𝐷‘𝑘) = (𝐷‘𝑗))
963962feq1d 6707 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑗 → ((𝐷‘𝑘):𝑋⟶ℂ ↔ (𝐷‘𝑗):𝑋⟶ℂ))
964266, 963imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)))
965961, 964, 583chvarfv 2229 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝐷‘𝑗):𝑋⟶ℂ)
966308, 958, 965vtocl 3543 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑖 + 1) − 𝑘) ∈ (0...𝑁)) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
967939, 955, 966syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (𝐷‘((𝑖 + 1) − 𝑘)):𝑋⟶ℂ)
968967, 937ffvelcdmd 7095 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) ∈ ℂ)
969938, 968mulcld 11265 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) ∈ ℂ)
970920, 969mulcld 11265 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...(𝑖 + 1))) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) ∈ ℂ)
971863, 685eleqtrrd 2832 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (ℤ≥‘0))
972 eluzfz2 13542 . . . . . . . . . . . . . . . . . . 19 ((𝑖 + 1) ∈ (ℤ≥‘0) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
973971, 972syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
974973ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (𝑖 + 1) ∈ (0...(𝑖 + 1)))
975 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑖 + 1) → ((𝑖 + 1)C𝑘) = ((𝑖 + 1)C(𝑖 + 1)))
976639fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑖 + 1) → ((𝐶‘𝑘)‘𝑥) = ((𝐶‘(𝑖 + 1))‘𝑥))
977 oveq2 7428 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑖 + 1) → ((𝑖 + 1) − 𝑘) = ((𝑖 + 1) − (𝑖 + 1)))
978977fveq2d 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑖 + 1) → (𝐷‘((𝑖 + 1) − 𝑘)) = (𝐷‘((𝑖 + 1) − (𝑖 + 1))))
979978fveq1d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑖 + 1) → ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥) = ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))
980976, 979oveq12d 7438 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑖 + 1) → (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)) = (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥)))
981975, 980oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑖 + 1) → (((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = (((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))))
982472, 912, 913, 970, 974, 981fsumsplit1 15724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))) = ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
983982eqcomd 2734 . . . . . . . . . . . . . . 15 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝑖 + 1)C(𝑖 + 1)) · (((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘((𝑖 + 1) − (𝑖 + 1)))‘𝑥))) + Σ𝑘 ∈ ((0...(𝑖 + 1)) ∖ {(𝑖 + 1)})(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
984886, 905, 9833eqtrd 2772 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → ((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + ((((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥)) + Σ𝑘 ∈ (1...𝑖)(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
985853, 862, 9843eqtrd 2772 . . . . . . . . . . . . 13 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (((((𝐶‘(𝑖 + 1))‘𝑥) · ((𝐷‘0)‘𝑥)) + (((𝐶‘0)‘𝑥) · ((𝐷‘(𝑖 + 1))‘𝑥))) + (Σ𝑗 ∈ (1...𝑖)((𝑖C(𝑗 − 1)) · (((𝐶‘𝑗)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑗))‘𝑥))) + Σ𝑘 ∈ (1...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
986776, 812, 9853eqtrd 2772 . . . . . . . . . . . 12 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → (Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))) + Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
987438, 442, 9863eqtrd 2772 . . . . . . . . . . 11 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))) = Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))
988987mpteq2dva 5248 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · ((((𝐶‘(𝑘 + 1))‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥)) + (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
989422, 988eqtrd 2768 . . . . . . . . 9 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
990989adantr 480 . . . . . . . 8 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
991189, 191, 9903eqtrd 2772 . . . . . . 7 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
992178, 179, 182, 991syl21anc 837 . . . . . 6 ((𝑖 ∈ (0..^𝑁) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))
9939923exp 1117 . . . . 5 (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑖) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑖)((𝑖C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑖 − 𝑘))‘𝑥))))) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘(𝑖 + 1)) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...(𝑖 + 1))(((𝑖 + 1)C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘((𝑖 + 1) − 𝑘))‘𝑥)))))))
99434, 47, 60, 73, 177, 993fzind2 13783 . . . 4 (𝑛 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑛 − 𝑘))‘𝑥))))))
99521, 994vtoclg 3540 . . 3 (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))))
9962, 6, 995sylc 65 . 2 (𝜑 → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥))))))
9971, 996mpd 15 1 (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1532   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  Vcvv 3471   ∖ cdif 3944   ⊆ wss 3947  𝒫 cpw 4603  {csn 4629  {cpr 4631   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420   ↑pm cpm 8846  Fincfn 8964  ℂcc 11137  ℝcr 11138  0cc0 11139  1c1 11140   + caddc 11142   · cmul 11144   < clt 11279   ≤ cle 11280   − cmin 11475  ℕcn 12243  ℕ0cn0 12503  ℤcz 12589  ℤ≥cuz 12853  ℝ+crp 13007  ...cfz 13517  ..^cfzo 13660  Ccbc 14294  Σcsu 15665   ↾t crest 17402  TopOpenctopn 17403  ℂfldccnfld 21279   D cdv 25805   D𝑛 cdvn 25806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217  ax-addf 11218
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9387  df-fi 9435  df-sup 9466  df-inf 9467  df-oi 9534  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-icc 13364  df-fz 13518  df-fzo 13661  df-seq 14000  df-exp 14060  df-fac 14266  df-bc 14295  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-clim 15465  df-sum 15666  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-mulr 17247  df-starv 17248  df-sca 17249  df-vsca 17250  df-ip 17251  df-tset 17252  df-ple 17253  df-ds 17255  df-unif 17256  df-hom 17257  df-cco 17258  df-rest 17404  df-topn 17405  df-0g 17423  df-gsum 17424  df-topgen 17425  df-pt 17426  df-prds 17429  df-xrs 17484  df-qtop 17489  df-imas 17490  df-xps 17492  df-mre 17566  df-mrc 17567  df-acs 17569  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18741  df-mulg 19024  df-cntz 19268  df-cmn 19737  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-fbas 21276  df-fg 21277  df-cnfld 21280  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22862  df-cld 22936  df-ntr 22937  df-cls 22938  df-nei 23015  df-lp 23053  df-perf 23054  df-cn 23144  df-cnp 23145  df-haus 23232  df-tx 23479  df-hmeo 23672  df-fil 23763  df-fm 23855  df-flim 23856  df-flf 23857  df-xms 24239  df-ms 24240  df-tms 24241  df-cncf 24811  df-limc 25808  df-dv 25809  df-dvn 25810
This theorem is referenced by:  dvnprodlem2  45335
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