Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvnmptdivc Structured version   Visualization version   GIF version

Theorem dvnmptdivc 44169
Description: Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
dvnmptdivc.s (𝜑𝑆 ∈ {ℝ, ℂ})
dvnmptdivc.x (𝜑𝑋𝑆)
dvnmptdivc.a ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
dvnmptdivc.b ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
dvnmptdivc.dvn ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵))
dvnmptdivc.c (𝜑𝐶 ∈ ℂ)
dvnmptdivc.cne0 (𝜑𝐶 ≠ 0)
dvnmptdivc.8 (𝜑𝑀 ∈ ℕ0)
Assertion
Ref Expression
dvnmptdivc ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
Distinct variable groups:   𝐴,𝑛   𝑥,𝐶   𝑛,𝑀,𝑥   𝑆,𝑛,𝑥   𝑛,𝑋,𝑥   𝜑,𝑛,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑛)   𝐶(𝑛)

Proof of Theorem dvnmptdivc
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2 simpl 483 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝜑)
3 fveq2 6842 . . . . 5 (𝑘 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0))
4 csbeq1 3858 . . . . . . 7 (𝑘 = 0 → 𝑘 / 𝑛𝐵 = 0 / 𝑛𝐵)
54oveq1d 7372 . . . . . 6 (𝑘 = 0 → (𝑘 / 𝑛𝐵 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
65mpteq2dv 5207 . . . . 5 (𝑘 = 0 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
73, 6eqeq12d 2752 . . . 4 (𝑘 = 0 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶))))
87imbi2d 340 . . 3 (𝑘 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))))
9 fveq2 6842 . . . . 5 (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10 csbeq1 3858 . . . . . . 7 (𝑘 = 𝑗𝑘 / 𝑛𝐵 = 𝑗 / 𝑛𝐵)
1110oveq1d 7372 . . . . . 6 (𝑘 = 𝑗 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝑗 / 𝑛𝐵 / 𝐶))
1211mpteq2dv 5207 . . . . 5 (𝑘 = 𝑗 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
139, 12eqeq12d 2752 . . . 4 (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
1413imbi2d 340 . . 3 (𝑘 = 𝑗 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))))
15 fveq2 6842 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16 csbeq1 3858 . . . . . . 7 (𝑘 = (𝑗 + 1) → 𝑘 / 𝑛𝐵 = (𝑗 + 1) / 𝑛𝐵)
1716oveq1d 7372 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑘 / 𝑛𝐵 / 𝐶) = ((𝑗 + 1) / 𝑛𝐵 / 𝐶))
1817mpteq2dv 5207 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
1915, 18eqeq12d 2752 . . . 4 (𝑘 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶))))
2019imbi2d 340 . . 3 (𝑘 = (𝑗 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))))
21 fveq2 6842 . . . . 5 (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22 csbeq1a 3869 . . . . . . . . 9 (𝑛 = 𝑘𝐵 = 𝑘 / 𝑛𝐵)
2322equcoms 2023 . . . . . . . 8 (𝑘 = 𝑛𝐵 = 𝑘 / 𝑛𝐵)
2423eqcomd 2742 . . . . . . 7 (𝑘 = 𝑛𝑘 / 𝑛𝐵 = 𝐵)
2524oveq1d 7372 . . . . . 6 (𝑘 = 𝑛 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝐵 / 𝐶))
2625mpteq2dv 5207 . . . . 5 (𝑘 = 𝑛 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
2721, 26eqeq12d 2752 . . . 4 (𝑘 = 𝑛 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2827imbi2d 340 . . 3 (𝑘 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))))
29 dvnmptdivc.s . . . . . . 7 (𝜑𝑆 ∈ {ℝ, ℂ})
30 recnprss 25268 . . . . . . 7 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3129, 30syl 17 . . . . . 6 (𝜑𝑆 ⊆ ℂ)
32 cnex 11132 . . . . . . . 8 ℂ ∈ V
3332a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
34 dvnmptdivc.a . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
35 dvnmptdivc.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℂ)
3635adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)
37 dvnmptdivc.cne0 . . . . . . . . . 10 (𝜑𝐶 ≠ 0)
3837adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ≠ 0)
3934, 36, 38divcld 11931 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) ∈ ℂ)
4039fmpttd 7063 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
41 dvnmptdivc.x . . . . . . 7 (𝜑𝑋𝑆)
42 elpm2r 8783 . . . . . . 7 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
4333, 29, 40, 41, 42syl22anc 837 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
44 dvn0 25288 . . . . . 6 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
4531, 43, 44syl2anc 584 . . . . 5 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
46 id 22 . . . . . . . . . . . 12 (𝜑𝜑)
47 dvnmptdivc.8 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℕ0)
48 nn0uz 12805 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
4947, 48eleqtrdi 2848 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘0))
50 eluzfz1 13448 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
5149, 50syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ (0...𝑀))
52 nfv 1917 . . . . . . . . . . . . . 14 𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
53 nfcv 2907 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)
54 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑛𝑋
55 nfcsb1v 3880 . . . . . . . . . . . . . . . 16 𝑛0 / 𝑛𝐵
5654, 55nfmpt 5212 . . . . . . . . . . . . . . 15 𝑛(𝑥𝑋0 / 𝑛𝐵)
5753, 56nfeq 2920 . . . . . . . . . . . . . 14 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)
5852, 57nfim 1899 . . . . . . . . . . . . 13 𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
59 c0ex 11149 . . . . . . . . . . . . 13 0 ∈ V
60 eleq1 2825 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
6160anbi2d 629 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
62 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑛 = 0 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0))
63 csbeq1a 3869 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝐵 = 0 / 𝑛𝐵)
6463mpteq2dv 5207 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑥𝑋𝐵) = (𝑥𝑋0 / 𝑛𝐵))
6562, 64eqeq12d 2752 . . . . . . . . . . . . . 14 (𝑛 = 0 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)))
6661, 65imbi12d 344 . . . . . . . . . . . . 13 (𝑛 = 0 → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))))
67 dvnmptdivc.dvn . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵))
6858, 59, 66, 67vtoclf 3516 . . . . . . . . . . . 12 ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
6946, 51, 68syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7069fveq1d 6844 . . . . . . . . . 10 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
7170adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
72 simpr 485 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
73 simpl 483 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝜑)
7451adantr 481 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 0 ∈ (0...𝑀))
75 0re 11157 . . . . . . . . . . . 12 0 ∈ ℝ
76 nfcv 2907 . . . . . . . . . . . . 13 𝑛0
77 nfv 1917 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))
78 nfcv 2907 . . . . . . . . . . . . . . 15 𝑛
7955, 78nfel 2921 . . . . . . . . . . . . . 14 𝑛0 / 𝑛𝐵 ∈ ℂ
8077, 79nfim 1899 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
81603anbi3d 1442 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))))
8263eleq1d 2822 . . . . . . . . . . . . . 14 (𝑛 = 0 → (𝐵 ∈ ℂ ↔ 0 / 𝑛𝐵 ∈ ℂ))
8381, 82imbi12d 344 . . . . . . . . . . . . 13 (𝑛 = 0 → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)))
84 dvnmptdivc.b . . . . . . . . . . . . 13 ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
8576, 80, 83, 84vtoclgf 3523 . . . . . . . . . . . 12 (0 ∈ ℝ → ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ))
8675, 85ax-mp 5 . . . . . . . . . . 11 ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
8773, 72, 74, 86syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 ∈ ℂ)
88 eqid 2736 . . . . . . . . . . 11 (𝑥𝑋0 / 𝑛𝐵) = (𝑥𝑋0 / 𝑛𝐵)
8988fvmpt2 6959 . . . . . . . . . 10 ((𝑥𝑋0 / 𝑛𝐵 ∈ ℂ) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9072, 87, 89syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝑋) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9171, 90eqtr2d 2777 . . . . . . . 8 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 = (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥))
9234fmpttd 7063 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋𝐴):𝑋⟶ℂ)
93 elpm2r 8783 . . . . . . . . . . . 12 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋𝐴):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
9433, 29, 92, 41, 93syl22anc 837 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
95 dvn0 25288 . . . . . . . . . . 11 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9631, 94, 95syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9796fveq1d 6844 . . . . . . . . 9 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
9897adantr 481 . . . . . . . 8 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
99 eqid 2736 . . . . . . . . . 10 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
10099fvmpt2 6959 . . . . . . . . 9 ((𝑥𝑋𝐴 ∈ ℂ) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10172, 34, 100syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10291, 98, 1013eqtrrd 2781 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 = 0 / 𝑛𝐵)
103102oveq1d 7372 . . . . . 6 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
104103mpteq2dva 5205 . . . . 5 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
10545, 104eqtrd 2776 . . . 4 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
106105a1i 11 . . 3 (𝑀 ∈ (ℤ‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶))))
107 simp3 1138 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
108 simp1 1136 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
109 simpr 485 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
110 simpl 483 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
111109, 110mpd 15 . . . . . 6 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
1121113adant1 1130 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
11331ad2antrr 724 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
11443ad2antrr 724 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
115 elfzofz 13588 . . . . . . . 8 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
116 elfznn0 13534 . . . . . . . . 9 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
117116ad2antlr 725 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
118115, 117sylanl2 679 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119 dvnp1 25289 . . . . . . 7 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
120113, 114, 118, 119syl3anc 1371 . . . . . 6 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
121 oveq2 7365 . . . . . . 7 (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
122121adantl 482 . . . . . 6 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
12331adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
12443adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
125 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
126125, 116syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
127115, 126sylan2 593 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
128123, 124, 127, 119syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
129128adantr 481 . . . . . . . . 9 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
13029adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
131 simplr 767 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 ∈ (0...𝑀))
13246ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝜑)
133 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
134132, 133, 1313jca 1128 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋𝑗 ∈ (0...𝑀)))
135 nfcv 2907 . . . . . . . . . . . . . 14 𝑛𝑗
136 nfv 1917 . . . . . . . . . . . . . . 15 𝑛(𝜑𝑥𝑋𝑗 ∈ (0...𝑀))
137135nfcsb1 3879 . . . . . . . . . . . . . . . 16 𝑛𝑗 / 𝑛𝐵
138137, 78nfel 2921 . . . . . . . . . . . . . . 15 𝑛𝑗 / 𝑛𝐵 ∈ ℂ
139136, 138nfim 1899 . . . . . . . . . . . . . 14 𝑛((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)
140 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
1411403anbi3d 1442 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋𝑗 ∈ (0...𝑀))))
142 csbeq1a 3869 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗𝐵 = 𝑗 / 𝑛𝐵)
143142eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑛𝐵 ∈ ℂ))
144141, 143imbi12d 344 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)))
145135, 139, 144, 84vtoclgf 3523 . . . . . . . . . . . . 13 (𝑗 ∈ (0...𝑀) → ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ))
146131, 134, 145sylc 65 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
147115, 146sylanl2 679 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
148 fzofzp1 13669 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
149148ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) ∈ (0...𝑀))
150115, 132sylanl2 679 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝜑)
151 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
152150, 151, 1493jca 1128 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
153 nfcv 2907 . . . . . . . . . . . . 13 𝑛(𝑗 + 1)
154 nfv 1917 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
155153nfcsb1 3879 . . . . . . . . . . . . . . 15 𝑛(𝑗 + 1) / 𝑛𝐵
156155, 78nfel 2921 . . . . . . . . . . . . . 14 𝑛(𝑗 + 1) / 𝑛𝐵 ∈ ℂ
157154, 156nfim 1899 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
158 eleq1 2825 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
1591583anbi3d 1442 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
160 csbeq1a 3869 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → 𝐵 = (𝑗 + 1) / 𝑛𝐵)
161160eleq1d 2822 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
162159, 161imbi12d 344 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)))
163153, 157, 162, 84vtoclgf 3523 . . . . . . . . . . . 12 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
164149, 152, 163sylc 65 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
165 simpl 483 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝜑)
166115adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
167 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑛(𝜑𝑗 ∈ (0...𝑀))
168 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)
16954, 137nfmpt 5212 . . . . . . . . . . . . . . . . . 18 𝑛(𝑥𝑋𝑗 / 𝑛𝐵)
170168, 169nfeq 2920 . . . . . . . . . . . . . . . . 17 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)
171167, 170nfim 1899 . . . . . . . . . . . . . . . 16 𝑛((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
172140anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑗 ∈ (0...𝑀))))
173 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
174142mpteq2dv 5207 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝑗 / 𝑛𝐵))
175173, 174eqeq12d 2752 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)))
176172, 175imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))))
177171, 176, 67chvarfv 2233 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
178165, 166, 177syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
179178eqcomd 2742 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝑗 / 𝑛𝐵) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
180179oveq2d 7373 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
181165, 94syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
182 dvnp1 25289 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
183123, 181, 127, 182syl3anc 1371 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
184183eqcomd 2742 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
185148adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
186165, 185jca 512 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
187 nfv 1917 . . . . . . . . . . . . . . 15 𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
188 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1))
18954, 155nfmpt 5212 . . . . . . . . . . . . . . . 16 𝑛(𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
190188, 189nfeq 2920 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
191187, 190nfim 1899 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
192158anbi2d 629 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
193 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
194160mpteq2dv 5207 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝑥𝑋𝐵) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
195193, 194eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
196192, 195imbi12d 344 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))))
197153, 191, 196, 67vtoclgf 3523 . . . . . . . . . . . . 13 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
198185, 186, 197sylc 65 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
199180, 184, 1983eqtrd 2780 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
20035adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
20137adantr 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
202130, 147, 164, 199, 200, 201dvmptdivc 25329 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
203202adantr 481 . . . . . . . . 9 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
204129, 122, 2033eqtrd 2780 . . . . . . . 8 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
205204eqcomd 2742 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
206205, 120, 1223eqtrrd 2781 . . . . . 6 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
207120, 122, 2063eqtrd 2780 . . . . 5 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
208107, 108, 112, 207syl21anc 836 . . . 4 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
2092083exp 1119 . . 3 (𝑗 ∈ (0..^𝑀) → ((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))))
2108, 14, 20, 28, 106, 209fzind2 13690 . 2 (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2111, 2, 210sylc 65 1 ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  csb 3855  wss 3910  {cpr 4588  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  pm cpm 8766  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   / cdiv 11812  0cn0 12413  cuz 12763  ...cfz 13424  ..^cfzo 13567   D cdv 25227   D𝑛 cdvn 25228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-dvn 25232
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator