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Theorem dvnmptdivc 45953
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 484 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2 simpl 482 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝜑)
3 fveq2 6906 . . . . 5 (𝑘 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0))
4 csbeq1 3902 . . . . . . 7 (𝑘 = 0 → 𝑘 / 𝑛𝐵 = 0 / 𝑛𝐵)
54oveq1d 7446 . . . . . 6 (𝑘 = 0 → (𝑘 / 𝑛𝐵 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
65mpteq2dv 5244 . . . . 5 (𝑘 = 0 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
73, 6eqeq12d 2753 . . . 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 6906 . . . . 5 (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10 csbeq1 3902 . . . . . . 7 (𝑘 = 𝑗𝑘 / 𝑛𝐵 = 𝑗 / 𝑛𝐵)
1110oveq1d 7446 . . . . . 6 (𝑘 = 𝑗 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝑗 / 𝑛𝐵 / 𝐶))
1211mpteq2dv 5244 . . . . 5 (𝑘 = 𝑗 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
139, 12eqeq12d 2753 . . . 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 6906 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16 csbeq1 3902 . . . . . . 7 (𝑘 = (𝑗 + 1) → 𝑘 / 𝑛𝐵 = (𝑗 + 1) / 𝑛𝐵)
1716oveq1d 7446 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑘 / 𝑛𝐵 / 𝐶) = ((𝑗 + 1) / 𝑛𝐵 / 𝐶))
1817mpteq2dv 5244 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
1915, 18eqeq12d 2753 . . . 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 6906 . . . . 5 (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22 csbeq1a 3913 . . . . . . . . 9 (𝑛 = 𝑘𝐵 = 𝑘 / 𝑛𝐵)
2322equcoms 2019 . . . . . . . 8 (𝑘 = 𝑛𝐵 = 𝑘 / 𝑛𝐵)
2423eqcomd 2743 . . . . . . 7 (𝑘 = 𝑛𝑘 / 𝑛𝐵 = 𝐵)
2524oveq1d 7446 . . . . . 6 (𝑘 = 𝑛 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝐵 / 𝐶))
2625mpteq2dv 5244 . . . . 5 (𝑘 = 𝑛 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
2721, 26eqeq12d 2753 . . . 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 25939 . . . . . . 7 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3129, 30syl 17 . . . . . 6 (𝜑𝑆 ⊆ ℂ)
32 cnex 11236 . . . . . . . 8 ℂ ∈ V
3332a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
34 dvnmptdivc.a . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
35 dvnmptdivc.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℂ)
3635adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)
37 dvnmptdivc.cne0 . . . . . . . . . 10 (𝜑𝐶 ≠ 0)
3837adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ≠ 0)
3934, 36, 38divcld 12043 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) ∈ ℂ)
4039fmpttd 7135 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
41 dvnmptdivc.x . . . . . . 7 (𝜑𝑋𝑆)
42 elpm2r 8885 . . . . . . 7 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
4333, 29, 40, 41, 42syl22anc 839 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
44 dvn0 25960 . . . . . 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 12920 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
4947, 48eleqtrdi 2851 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘0))
50 eluzfz1 13571 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
5149, 50syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ (0...𝑀))
52 nfv 1914 . . . . . . . . . . . . . 14 𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
53 nfcv 2905 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)
54 nfcv 2905 . . . . . . . . . . . . . . . 16 𝑛𝑋
55 nfcsb1v 3923 . . . . . . . . . . . . . . . 16 𝑛0 / 𝑛𝐵
5654, 55nfmpt 5249 . . . . . . . . . . . . . . 15 𝑛(𝑥𝑋0 / 𝑛𝐵)
5753, 56nfeq 2919 . . . . . . . . . . . . . 14 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)
5852, 57nfim 1896 . . . . . . . . . . . . 13 𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
59 c0ex 11255 . . . . . . . . . . . . 13 0 ∈ V
60 eleq1 2829 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
6160anbi2d 630 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
62 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑛 = 0 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0))
63 csbeq1a 3913 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝐵 = 0 / 𝑛𝐵)
6463mpteq2dv 5244 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑥𝑋𝐵) = (𝑥𝑋0 / 𝑛𝐵))
6562, 64eqeq12d 2753 . . . . . . . . . . . . . 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 3564 . . . . . . . . . . . 12 ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
6946, 51, 68syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7069fveq1d 6908 . . . . . . . . . 10 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
7170adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
72 simpr 484 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
73 simpl 482 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝜑)
7451adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 0 ∈ (0...𝑀))
75 0re 11263 . . . . . . . . . . . 12 0 ∈ ℝ
76 nfcv 2905 . . . . . . . . . . . . 13 𝑛0
77 nfv 1914 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))
78 nfcv 2905 . . . . . . . . . . . . . . 15 𝑛
7955, 78nfel 2920 . . . . . . . . . . . . . 14 𝑛0 / 𝑛𝐵 ∈ ℂ
8077, 79nfim 1896 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
81603anbi3d 1444 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))))
8263eleq1d 2826 . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . 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 1373 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 ∈ ℂ)
88 eqid 2737 . . . . . . . . . . 11 (𝑥𝑋0 / 𝑛𝐵) = (𝑥𝑋0 / 𝑛𝐵)
8988fvmpt2 7027 . . . . . . . . . 10 ((𝑥𝑋0 / 𝑛𝐵 ∈ ℂ) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9072, 87, 89syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝑋) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9171, 90eqtr2d 2778 . . . . . . . 8 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 = (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥))
9234fmpttd 7135 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋𝐴):𝑋⟶ℂ)
93 elpm2r 8885 . . . . . . . . . . . 12 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋𝐴):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
9433, 29, 92, 41, 93syl22anc 839 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
95 dvn0 25960 . . . . . . . . . . 11 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9631, 94, 95syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9796fveq1d 6908 . . . . . . . . 9 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
9897adantr 480 . . . . . . . 8 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
99 eqid 2737 . . . . . . . . . 10 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
10099fvmpt2 7027 . . . . . . . . 9 ((𝑥𝑋𝐴 ∈ ℂ) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10172, 34, 100syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10291, 98, 1013eqtrrd 2782 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 = 0 / 𝑛𝐵)
103102oveq1d 7446 . . . . . 6 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
104103mpteq2dva 5242 . . . . 5 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
10545, 104eqtrd 2777 . . . 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 1139 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
108 simp1 1137 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
109 simpr 484 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
110 simpl 482 . . . . . . 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 1131 . . . . 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 726 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
11443ad2antrr 726 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
115 elfzofz 13715 . . . . . . . 8 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
116 elfznn0 13660 . . . . . . . . 9 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
117116ad2antlr 727 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
118115, 117sylanl2 681 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119 dvnp1 25961 . . . . . . 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 1373 . . . . . 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 7439 . . . . . . 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 481 . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
12443adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
125 simpr 484 . . . . . . . . . . . . 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 1373 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
129128adantr 480 . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
131 simplr 769 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 ∈ (0...𝑀))
13246ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝜑)
133 simpr 484 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
134132, 133, 1313jca 1129 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋𝑗 ∈ (0...𝑀)))
135 nfcv 2905 . . . . . . . . . . . . . 14 𝑛𝑗
136 nfv 1914 . . . . . . . . . . . . . . 15 𝑛(𝜑𝑥𝑋𝑗 ∈ (0...𝑀))
137135nfcsb1 3922 . . . . . . . . . . . . . . . 16 𝑛𝑗 / 𝑛𝐵
138137, 78nfel 2920 . . . . . . . . . . . . . . 15 𝑛𝑗 / 𝑛𝐵 ∈ ℂ
139136, 138nfim 1896 . . . . . . . . . . . . . 14 𝑛((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)
140 eleq1 2829 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
1411403anbi3d 1444 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋𝑗 ∈ (0...𝑀))))
142 csbeq1a 3913 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗𝐵 = 𝑗 / 𝑛𝐵)
143142eleq1d 2826 . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . 13 (𝑗 ∈ (0...𝑀) → ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ))
146131, 134, 145sylc 65 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
147115, 146sylanl2 681 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
148 fzofzp1 13803 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
149148ad2antlr 727 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) ∈ (0...𝑀))
150115, 132sylanl2 681 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝜑)
151 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
152150, 151, 1493jca 1129 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
153 nfcv 2905 . . . . . . . . . . . . 13 𝑛(𝑗 + 1)
154 nfv 1914 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
155153nfcsb1 3922 . . . . . . . . . . . . . . 15 𝑛(𝑗 + 1) / 𝑛𝐵
156155, 78nfel 2920 . . . . . . . . . . . . . 14 𝑛(𝑗 + 1) / 𝑛𝐵 ∈ ℂ
157154, 156nfim 1896 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
158 eleq1 2829 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
1591583anbi3d 1444 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
160 csbeq1a 3913 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → 𝐵 = (𝑗 + 1) / 𝑛𝐵)
161160eleq1d 2826 . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝜑)
166115adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
167 nfv 1914 . . . . . . . . . . . . . . . . 17 𝑛(𝜑𝑗 ∈ (0...𝑀))
168 nfcv 2905 . . . . . . . . . . . . . . . . . 18 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)
16954, 137nfmpt 5249 . . . . . . . . . . . . . . . . . 18 𝑛(𝑥𝑋𝑗 / 𝑛𝐵)
170168, 169nfeq 2919 . . . . . . . . . . . . . . . . 17 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)
171167, 170nfim 1896 . . . . . . . . . . . . . . . 16 𝑛((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
172140anbi2d 630 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑗 ∈ (0...𝑀))))
173 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
174142mpteq2dv 5244 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝑗 / 𝑛𝐵))
175173, 174eqeq12d 2753 . . . . . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝑗 / 𝑛𝐵) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
180179oveq2d 7447 . . . . . . . . . . . 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 25961 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
183123, 181, 127, 182syl3anc 1373 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
184183eqcomd 2743 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
185148adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
186165, 185jca 511 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
187 nfv 1914 . . . . . . . . . . . . . . 15 𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
188 nfcv 2905 . . . . . . . . . . . . . . . 16 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1))
18954, 155nfmpt 5249 . . . . . . . . . . . . . . . 16 𝑛(𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
190188, 189nfeq 2919 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
191187, 190nfim 1896 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
192158anbi2d 630 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
193 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
194160mpteq2dv 5244 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝑥𝑋𝐵) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
195193, 194eqeq12d 2753 . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . 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 2781 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
20035adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
20137adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
202130, 147, 164, 199, 200, 201dvmptdivc 26003 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
203202adantr 480 . . . . . . . . 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 2781 . . . . . . . 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 2743 . . . . . . 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 2782 . . . . . 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 2781 . . . . 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 838 . . . 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 1120 . . 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 13824 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  csb 3899  wss 3951  {cpr 4628  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  pm cpm 8867  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   / cdiv 11920  0cn0 12526  cuz 12878  ...cfz 13547  ..^cfzo 13694   D cdv 25898   D𝑛 cdvn 25899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-dvn 25903
This theorem is referenced by: (None)
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