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Theorem dvnmptdivc 40471
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 476 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2 simpl 472 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝜑)
3 fveq2 6229 . . . . 5 (𝑘 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0))
4 csbeq1 3569 . . . . . . 7 (𝑘 = 0 → 𝑘 / 𝑛𝐵 = 0 / 𝑛𝐵)
54oveq1d 6705 . . . . . 6 (𝑘 = 0 → (𝑘 / 𝑛𝐵 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
65mpteq2dv 4778 . . . . 5 (𝑘 = 0 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
73, 6eqeq12d 2666 . . . 4 (𝑘 = 0 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶))))
87imbi2d 329 . . 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 6229 . . . . 5 (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10 csbeq1 3569 . . . . . . 7 (𝑘 = 𝑗𝑘 / 𝑛𝐵 = 𝑗 / 𝑛𝐵)
1110oveq1d 6705 . . . . . 6 (𝑘 = 𝑗 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝑗 / 𝑛𝐵 / 𝐶))
1211mpteq2dv 4778 . . . . 5 (𝑘 = 𝑗 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
139, 12eqeq12d 2666 . . . 4 (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
1413imbi2d 329 . . 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 6229 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16 csbeq1 3569 . . . . . . 7 (𝑘 = (𝑗 + 1) → 𝑘 / 𝑛𝐵 = (𝑗 + 1) / 𝑛𝐵)
1716oveq1d 6705 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑘 / 𝑛𝐵 / 𝐶) = ((𝑗 + 1) / 𝑛𝐵 / 𝐶))
1817mpteq2dv 4778 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
1915, 18eqeq12d 2666 . . . 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 329 . . 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 6229 . . . . 5 (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22 csbeq1a 3575 . . . . . . . . 9 (𝑛 = 𝑘𝐵 = 𝑘 / 𝑛𝐵)
2322equcoms 1993 . . . . . . . 8 (𝑘 = 𝑛𝐵 = 𝑘 / 𝑛𝐵)
2423eqcomd 2657 . . . . . . 7 (𝑘 = 𝑛𝑘 / 𝑛𝐵 = 𝐵)
2524oveq1d 6705 . . . . . 6 (𝑘 = 𝑛 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝐵 / 𝐶))
2625mpteq2dv 4778 . . . . 5 (𝑘 = 𝑛 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
2721, 26eqeq12d 2666 . . . 4 (𝑘 = 𝑛 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2827imbi2d 329 . . 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 23713 . . . . . . 7 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3129, 30syl 17 . . . . . 6 (𝜑𝑆 ⊆ ℂ)
32 cnex 10055 . . . . . . . 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 10839 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40 eqid 2651 . . . . . . . 8 (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (𝐴 / 𝐶))
4139, 40fmptd 6425 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
42 dvnmptdivc.x . . . . . . 7 (𝜑𝑋𝑆)
43 elpm2r 7917 . . . . . . 7 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
4433, 29, 41, 42, 43syl22anc 1367 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
45 dvn0 23732 . . . . . 6 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
4631, 44, 45syl2anc 694 . . . . 5 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
47 id 22 . . . . . . . . . . . 12 (𝜑𝜑)
48 dvnmptdivc.8 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℕ0)
49 nn0uz 11760 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
5048, 49syl6eleq 2740 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘0))
51 eluzfz1 12386 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
5250, 51syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ (0...𝑀))
53 nfv 1883 . . . . . . . . . . . . . 14 𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
54 nfcv 2793 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)
55 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑛𝑋
56 nfcsb1v 3582 . . . . . . . . . . . . . . . 16 𝑛0 / 𝑛𝐵
5755, 56nfmpt 4779 . . . . . . . . . . . . . . 15 𝑛(𝑥𝑋0 / 𝑛𝐵)
5854, 57nfeq 2805 . . . . . . . . . . . . . 14 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)
5953, 58nfim 1865 . . . . . . . . . . . . 13 𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
60 c0ex 10072 . . . . . . . . . . . . 13 0 ∈ V
61 eleq1 2718 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
6261anbi2d 740 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
63 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑛 = 0 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0))
64 csbeq1a 3575 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝐵 = 0 / 𝑛𝐵)
6564mpteq2dv 4778 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑥𝑋𝐵) = (𝑥𝑋0 / 𝑛𝐵))
6663, 65eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑛 = 0 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)))
6762, 66imbi12d 333 . . . . . . . . . . . . 13 (𝑛 = 0 → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))))
68 dvnmptdivc.dvn . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵))
6959, 60, 67, 68vtoclf 3289 . . . . . . . . . . . 12 ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7047, 52, 69syl2anc 694 . . . . . . . . . . 11 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7170fveq1d 6231 . . . . . . . . . 10 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
7271adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
73 simpr 476 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
74 simpl 472 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝜑)
7552adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 0 ∈ (0...𝑀))
76 0re 10078 . . . . . . . . . . . 12 0 ∈ ℝ
77 nfcv 2793 . . . . . . . . . . . . 13 𝑛0
78 nfv 1883 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))
79 nfcv 2793 . . . . . . . . . . . . . . 15 𝑛
8056, 79nfel 2806 . . . . . . . . . . . . . 14 𝑛0 / 𝑛𝐵 ∈ ℂ
8178, 80nfim 1865 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
82613anbi3d 1445 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))))
8364eleq1d 2715 . . . . . . . . . . . . . 14 (𝑛 = 0 → (𝐵 ∈ ℂ ↔ 0 / 𝑛𝐵 ∈ ℂ))
8482, 83imbi12d 333 . . . . . . . . . . . . 13 (𝑛 = 0 → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)))
85 dvnmptdivc.b . . . . . . . . . . . . 13 ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
8677, 81, 84, 85vtoclgf 3295 . . . . . . . . . . . 12 (0 ∈ ℝ → ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ))
8776, 86ax-mp 5 . . . . . . . . . . 11 ((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
8874, 73, 75, 87syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 ∈ ℂ)
89 eqid 2651 . . . . . . . . . . 11 (𝑥𝑋0 / 𝑛𝐵) = (𝑥𝑋0 / 𝑛𝐵)
9089fvmpt2 6330 . . . . . . . . . 10 ((𝑥𝑋0 / 𝑛𝐵 ∈ ℂ) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9173, 88, 90syl2anc 694 . . . . . . . . 9 ((𝜑𝑥𝑋) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9272, 91eqtr2d 2686 . . . . . . . 8 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 = (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥))
93 eqid 2651 . . . . . . . . . . . . 13 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
9434, 93fmptd 6425 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋𝐴):𝑋⟶ℂ)
95 elpm2r 7917 . . . . . . . . . . . 12 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋𝐴):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
9633, 29, 94, 42, 95syl22anc 1367 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
97 dvn0 23732 . . . . . . . . . . 11 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9831, 96, 97syl2anc 694 . . . . . . . . . 10 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9998fveq1d 6231 . . . . . . . . 9 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
10099adantr 480 . . . . . . . 8 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
10193fvmpt2 6330 . . . . . . . . 9 ((𝑥𝑋𝐴 ∈ ℂ) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10273, 34, 101syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10392, 100, 1023eqtrrd 2690 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 = 0 / 𝑛𝐵)
104103oveq1d 6705 . . . . . 6 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
105104mpteq2dva 4777 . . . . 5 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
10646, 105eqtrd 2685 . . . 4 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
107106a1i 11 . . 3 (𝑀 ∈ (ℤ‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶))))
108 simp3 1083 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
109 simp1 1081 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
110 simpr 476 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
111 simpl 472 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
112110, 111mpd 15 . . . . . 6 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
1131123adant1 1099 . . . . 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)))
11431ad2antrr 762 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
11544ad2antrr 762 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
116 elfzofz 12524 . . . . . . . 8 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
117 elfznn0 12471 . . . . . . . . 9 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
118117ad2antlr 763 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119116, 118sylanl2 684 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
120 dvnp1 23733 . . . . . . 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)))
121114, 115, 119, 120syl3anc 1366 . . . . . 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)))
122 oveq2 6698 . . . . . . 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))))
123122adantl 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))))
12431adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
12544adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
126 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
127126, 117syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
128116, 127sylan2 490 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
129124, 125, 128, 120syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
130129adantr 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)))
13129adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
132 simplr 807 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 ∈ (0...𝑀))
13347ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝜑)
134 simpr 476 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
135133, 134, 1323jca 1261 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋𝑗 ∈ (0...𝑀)))
136 nfcv 2793 . . . . . . . . . . . . . 14 𝑛𝑗
137 nfv 1883 . . . . . . . . . . . . . . 15 𝑛(𝜑𝑥𝑋𝑗 ∈ (0...𝑀))
138136nfcsb1 3581 . . . . . . . . . . . . . . . 16 𝑛𝑗 / 𝑛𝐵
139138, 79nfel 2806 . . . . . . . . . . . . . . 15 𝑛𝑗 / 𝑛𝐵 ∈ ℂ
140137, 139nfim 1865 . . . . . . . . . . . . . 14 𝑛((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)
141 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
1421413anbi3d 1445 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋𝑗 ∈ (0...𝑀))))
143 csbeq1a 3575 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗𝐵 = 𝑗 / 𝑛𝐵)
144143eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑛𝐵 ∈ ℂ))
145142, 144imbi12d 333 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)))
146136, 140, 145, 85vtoclgf 3295 . . . . . . . . . . . . 13 (𝑗 ∈ (0...𝑀) → ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ))
147132, 135, 146sylc 65 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
148116, 147sylanl2 684 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
149 fzofzp1 12605 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
150149ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) ∈ (0...𝑀))
151116, 133sylanl2 684 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝜑)
152 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
153151, 152, 1503jca 1261 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
154 nfcv 2793 . . . . . . . . . . . . 13 𝑛(𝑗 + 1)
155 nfv 1883 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
156154nfcsb1 3581 . . . . . . . . . . . . . . 15 𝑛(𝑗 + 1) / 𝑛𝐵
157156, 79nfel 2806 . . . . . . . . . . . . . 14 𝑛(𝑗 + 1) / 𝑛𝐵 ∈ ℂ
158155, 157nfim 1865 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
159 eleq1 2718 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
1601593anbi3d 1445 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
161 csbeq1a 3575 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → 𝐵 = (𝑗 + 1) / 𝑛𝐵)
162161eleq1d 2715 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
163160, 162imbi12d 333 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)))
164154, 158, 163, 85vtoclgf 3295 . . . . . . . . . . . 12 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
165150, 153, 164sylc 65 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
166 simpl 472 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝜑)
167116adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
168 nfv 1883 . . . . . . . . . . . . . . . . 17 𝑛(𝜑𝑗 ∈ (0...𝑀))
169 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)
17055, 138nfmpt 4779 . . . . . . . . . . . . . . . . . 18 𝑛(𝑥𝑋𝑗 / 𝑛𝐵)
171169, 170nfeq 2805 . . . . . . . . . . . . . . . . 17 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)
172168, 171nfim 1865 . . . . . . . . . . . . . . . 16 𝑛((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
173141anbi2d 740 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑗 ∈ (0...𝑀))))
174 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
175143mpteq2dv 4778 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝑗 / 𝑛𝐵))
176174, 175eqeq12d 2666 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)))
177173, 176imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))))
178172, 177, 68chvar 2298 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
179166, 167, 178syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
180179eqcomd 2657 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝑗 / 𝑛𝐵) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
181180oveq2d 6706 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
182166, 96syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
183 dvnp1 23733 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
184124, 182, 128, 183syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
185184eqcomd 2657 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
186149adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
187166, 186jca 553 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
188 nfv 1883 . . . . . . . . . . . . . . 15 𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
189 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1))
19055, 156nfmpt 4779 . . . . . . . . . . . . . . . 16 𝑛(𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
191189, 190nfeq 2805 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
192188, 191nfim 1865 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
193159anbi2d 740 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
194 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
195161mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝑥𝑋𝐵) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
196194, 195eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
197193, 196imbi12d 333 . . . . . . . . . . . . . 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))))
198154, 192, 197, 68vtoclgf 3295 . . . . . . . . . . . . 13 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
199186, 187, 198sylc 65 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
200181, 185, 1993eqtrd 2689 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
20135adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
20237adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
203131, 148, 165, 200, 201, 202dvmptdivc 23773 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
204203adantr 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)))
205130, 123, 2043eqtrd 2689 . . . . . . . 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)))
206205eqcomd 2657 . . . . . . 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)))
207206, 121, 1233eqtrrd 2690 . . . . . 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)))
208121, 123, 2073eqtrd 2689 . . . . 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)))
209108, 109, 113, 208syl21anc 1365 . . . 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)))
2102093exp 1283 . . 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)))))
2118, 14, 20, 28, 107, 210fzind2 12626 . 2 (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2121, 2, 211sylc 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  csb 3566  wss 3607  {cpr 4212  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  pm cpm 7900  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   / cdiv 10722  0cn0 11330  cuz 11725  ...cfz 12364  ..^cfzo 12504   D cdv 23672   D𝑛 cdvn 23673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-dvn 23677
This theorem is referenced by: (None)
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