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Theorem dvnmptdivc 38652
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 475 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2 simpl 471 . 2 ((𝜑𝑛 ∈ (0...𝑀)) → 𝜑)
3 fveq2 6088 . . . . 5 (𝑘 = 0 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0))
4 csbeq1 3501 . . . . . . 7 (𝑘 = 0 → 𝑘 / 𝑛𝐵 = 0 / 𝑛𝐵)
54oveq1d 6542 . . . . . 6 (𝑘 = 0 → (𝑘 / 𝑛𝐵 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
65mpteq2dv 4667 . . . . 5 (𝑘 = 0 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
73, 6eqeq12d 2624 . . . 4 (𝑘 = 0 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶))))
87imbi2d 328 . . 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 6088 . . . . 5 (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10 csbeq1 3501 . . . . . . 7 (𝑘 = 𝑗𝑘 / 𝑛𝐵 = 𝑗 / 𝑛𝐵)
1110oveq1d 6542 . . . . . 6 (𝑘 = 𝑗 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝑗 / 𝑛𝐵 / 𝐶))
1211mpteq2dv 4667 . . . . 5 (𝑘 = 𝑗 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶)))
139, 12eqeq12d 2624 . . . 4 (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))))
1413imbi2d 328 . . 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 6088 . . . . 5 (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16 csbeq1 3501 . . . . . . 7 (𝑘 = (𝑗 + 1) → 𝑘 / 𝑛𝐵 = (𝑗 + 1) / 𝑛𝐵)
1716oveq1d 6542 . . . . . 6 (𝑘 = (𝑗 + 1) → (𝑘 / 𝑛𝐵 / 𝐶) = ((𝑗 + 1) / 𝑛𝐵 / 𝐶))
1817mpteq2dv 4667 . . . . 5 (𝑘 = (𝑗 + 1) → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
1915, 18eqeq12d 2624 . . . 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 328 . . 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 6088 . . . . 5 (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22 csbeq1a 3507 . . . . . . . . 9 (𝑛 = 𝑘𝐵 = 𝑘 / 𝑛𝐵)
2322equcoms 1933 . . . . . . . 8 (𝑘 = 𝑛𝐵 = 𝑘 / 𝑛𝐵)
2423eqcomd 2615 . . . . . . 7 (𝑘 = 𝑛𝑘 / 𝑛𝐵 = 𝐵)
2524oveq1d 6542 . . . . . 6 (𝑘 = 𝑛 → (𝑘 / 𝑛𝐵 / 𝐶) = (𝐵 / 𝐶))
2625mpteq2dv 4667 . . . . 5 (𝑘 = 𝑛 → (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
2721, 26eqeq12d 2624 . . . 4 (𝑘 = 𝑛 → (((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥𝑋 ↦ (𝑘 / 𝑛𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2827imbi2d 328 . . 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 23419 . . . . . . 7 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3129, 30syl 17 . . . . . 6 (𝜑𝑆 ⊆ ℂ)
32 cnex 9874 . . . . . . . 8 ℂ ∈ V
3332a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
34 dvnmptdivc.a . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)
35 dvnmptdivc.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℂ)
3635adantr 479 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)
37 dvnmptdivc.cne0 . . . . . . . . . 10 (𝜑𝐶 ≠ 0)
3837adantr 479 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐶 ≠ 0)
3934, 36, 38divcld 10653 . . . . . . . 8 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40 eqid 2609 . . . . . . . 8 (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (𝐴 / 𝐶))
4139, 40fmptd 6277 . . . . . . 7 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
42 dvnmptdivc.x . . . . . . 7 (𝜑𝑋𝑆)
43 elpm2r 7739 . . . . . . 7 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
4433, 29, 41, 42, 43syl22anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
45 dvn0 23438 . . . . . 6 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
4631, 44, 45syl2anc 690 . . . . 5 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥𝑋 ↦ (𝐴 / 𝐶)))
47 id 22 . . . . . . . . . . . 12 (𝜑𝜑)
48 dvnmptdivc.8 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℕ0)
49 nn0uz 11557 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
5048, 49syl6eleq 2697 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (ℤ‘0))
51 eluzfz1 12177 . . . . . . . . . . . . 13 (𝑀 ∈ (ℤ‘0) → 0 ∈ (0...𝑀))
5250, 51syl 17 . . . . . . . . . . . 12 (𝜑 → 0 ∈ (0...𝑀))
53 nfv 1829 . . . . . . . . . . . . . 14 𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
54 nfcv 2750 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)
55 nfcv 2750 . . . . . . . . . . . . . . . 16 𝑛𝑋
56 nfcsb1v 3514 . . . . . . . . . . . . . . . 16 𝑛0 / 𝑛𝐵
5755, 56nfmpt 4668 . . . . . . . . . . . . . . 15 𝑛(𝑥𝑋0 / 𝑛𝐵)
5854, 57nfeq 2761 . . . . . . . . . . . . . 14 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)
5953, 58nfim 1812 . . . . . . . . . . . . 13 𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
60 c0ex 9891 . . . . . . . . . . . . 13 0 ∈ V
61 eleq1 2675 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
6261anbi2d 735 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
63 fveq2 6088 . . . . . . . . . . . . . . 15 (𝑛 = 0 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0))
64 csbeq1a 3507 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → 𝐵 = 0 / 𝑛𝐵)
6564mpteq2dv 4667 . . . . . . . . . . . . . . 15 (𝑛 = 0 → (𝑥𝑋𝐵) = (𝑥𝑋0 / 𝑛𝐵))
6663, 65eqeq12d 2624 . . . . . . . . . . . . . 14 (𝑛 = 0 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵)))
6762, 66imbi12d 332 . . . . . . . . . . . . 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 3230 . . . . . . . . . . . 12 ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7047, 52, 69syl2anc 690 . . . . . . . . . . 11 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋0 / 𝑛𝐵))
7170fveq1d 6090 . . . . . . . . . 10 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
7271adantr 479 . . . . . . . . 9 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋0 / 𝑛𝐵)‘𝑥))
73 simpr 475 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
74 simpl 471 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 𝜑)
7552adantr 479 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → 0 ∈ (0...𝑀))
76 0re 9897 . . . . . . . . . . . 12 0 ∈ ℝ
77 nfcv 2750 . . . . . . . . . . . . 13 𝑛0
78 nfv 1829 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))
79 nfcv 2750 . . . . . . . . . . . . . . 15 𝑛
8056, 79nfel 2762 . . . . . . . . . . . . . 14 𝑛0 / 𝑛𝐵 ∈ ℂ
8178, 80nfim 1812 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀)) → 0 / 𝑛𝐵 ∈ ℂ)
82613anbi3d 1396 . . . . . . . . . . . . . 14 (𝑛 = 0 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ 0 ∈ (0...𝑀))))
8364eleq1d 2671 . . . . . . . . . . . . . 14 (𝑛 = 0 → (𝐵 ∈ ℂ ↔ 0 / 𝑛𝐵 ∈ ℂ))
8482, 83imbi12d 332 . . . . . . . . . . . . 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 3236 . . . . . . . . . . . 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 1317 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 ∈ ℂ)
89 eqid 2609 . . . . . . . . . . 11 (𝑥𝑋0 / 𝑛𝐵) = (𝑥𝑋0 / 𝑛𝐵)
9089fvmpt2 6185 . . . . . . . . . 10 ((𝑥𝑋0 / 𝑛𝐵 ∈ ℂ) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9173, 88, 90syl2anc 690 . . . . . . . . 9 ((𝜑𝑥𝑋) → ((𝑥𝑋0 / 𝑛𝐵)‘𝑥) = 0 / 𝑛𝐵)
9272, 91eqtr2d 2644 . . . . . . . 8 ((𝜑𝑥𝑋) → 0 / 𝑛𝐵 = (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥))
93 eqid 2609 . . . . . . . . . . . . 13 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
9434, 93fmptd 6277 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋𝐴):𝑋⟶ℂ)
95 elpm2r 7739 . . . . . . . . . . . 12 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥𝑋𝐴):𝑋⟶ℂ ∧ 𝑋𝑆)) → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
9633, 29, 94, 42, 95syl22anc 1318 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆))
97 dvn0 23438 . . . . . . . . . . 11 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9831, 96, 97syl2anc 690 . . . . . . . . . 10 (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘0) = (𝑥𝑋𝐴))
9998fveq1d 6090 . . . . . . . . 9 (𝜑 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
10099adantr 479 . . . . . . . 8 ((𝜑𝑥𝑋) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘0)‘𝑥) = ((𝑥𝑋𝐴)‘𝑥))
10193fvmpt2 6185 . . . . . . . . 9 ((𝑥𝑋𝐴 ∈ ℂ) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10273, 34, 101syl2anc 690 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
10392, 100, 1023eqtrrd 2648 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴 = 0 / 𝑛𝐵)
104103oveq1d 6542 . . . . . 6 ((𝜑𝑥𝑋) → (𝐴 / 𝐶) = (0 / 𝑛𝐵 / 𝐶))
105104mpteq2dva 4666 . . . . 5 (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐶)) = (𝑥𝑋 ↦ (0 / 𝑛𝐵 / 𝐶)))
10646, 105eqtrd 2643 . . . 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 1055 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
109 simp1 1053 . . . . 5 ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
110 simpr 475 . . . . . . 7 (((𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
111 simpl 471 . . . . . . 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 1071 . . . . 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 757 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
11544ad2antrr 757 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
116 elfzofz 12312 . . . . . . . 8 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
117 elfznn0 12260 . . . . . . . . 9 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
118117ad2antlr 758 . . . . . . . 8 (((𝜑𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119116, 118sylanl2 680 . . . . . . 7 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
120 dvnp1 23439 . . . . . . 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 1317 . . . . . 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 6535 . . . . . . 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 480 . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
12544adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
126 simpr 475 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
127126, 117syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
128116, 127sylan2 489 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
129124, 125, 128, 120syl3anc 1317 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
130129adantr 479 . . . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
132 simplr 787 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 ∈ (0...𝑀))
13347ad2antrr 757 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝜑)
134 simpr 475 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
135133, 134, 1323jca 1234 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋𝑗 ∈ (0...𝑀)))
136 nfcv 2750 . . . . . . . . . . . . . 14 𝑛𝑗
137 nfv 1829 . . . . . . . . . . . . . . 15 𝑛(𝜑𝑥𝑋𝑗 ∈ (0...𝑀))
138136nfcsb1 3513 . . . . . . . . . . . . . . . 16 𝑛𝑗 / 𝑛𝐵
139138, 79nfel 2762 . . . . . . . . . . . . . . 15 𝑛𝑗 / 𝑛𝐵 ∈ ℂ
140137, 139nfim 1812 . . . . . . . . . . . . . 14 𝑛((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)
141 eleq1 2675 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
1421413anbi3d 1396 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋𝑗 ∈ (0...𝑀))))
143 csbeq1a 3507 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗𝐵 = 𝑗 / 𝑛𝐵)
144143eleq1d 2671 . . . . . . . . . . . . . . 15 (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝑗 / 𝑛𝐵 ∈ ℂ))
145142, 144imbi12d 332 . . . . . . . . . . . . . 14 (𝑛 = 𝑗 → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ)))
146136, 140, 145, 85vtoclgf 3236 . . . . . . . . . . . . 13 (𝑗 ∈ (0...𝑀) → ((𝜑𝑥𝑋𝑗 ∈ (0...𝑀)) → 𝑗 / 𝑛𝐵 ∈ ℂ))
147132, 135, 146sylc 62 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
148116, 147sylanl2 680 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑗 / 𝑛𝐵 ∈ ℂ)
149 fzofzp1 12389 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
150149ad2antlr 758 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) ∈ (0...𝑀))
151116, 133sylanl2 680 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝜑)
152 simpr 475 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → 𝑥𝑋)
153151, 152, 1503jca 1234 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
154 nfcv 2750 . . . . . . . . . . . . 13 𝑛(𝑗 + 1)
155 nfv 1829 . . . . . . . . . . . . . 14 𝑛(𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
156154nfcsb1 3513 . . . . . . . . . . . . . . 15 𝑛(𝑗 + 1) / 𝑛𝐵
157156, 79nfel 2762 . . . . . . . . . . . . . 14 𝑛(𝑗 + 1) / 𝑛𝐵 ∈ ℂ
158155, 157nfim 1812 . . . . . . . . . . . . 13 𝑛((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
159 eleq1 2675 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
1601593anbi3d 1396 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
161 csbeq1a 3507 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → 𝐵 = (𝑗 + 1) / 𝑛𝐵)
162161eleq1d 2671 . . . . . . . . . . . . . 14 (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
163160, 162imbi12d 332 . . . . . . . . . . . . 13 (𝑛 = (𝑗 + 1) → (((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)))
164154, 158, 163, 85vtoclgf 3236 . . . . . . . . . . . 12 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑𝑥𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ))
165150, 153, 164sylc 62 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (0..^𝑀)) ∧ 𝑥𝑋) → (𝑗 + 1) / 𝑛𝐵 ∈ ℂ)
166 simpl 471 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝜑)
167116adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
168 nfv 1829 . . . . . . . . . . . . . . . . 17 𝑛(𝜑𝑗 ∈ (0...𝑀))
169 nfcv 2750 . . . . . . . . . . . . . . . . . 18 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)
17055, 138nfmpt 4668 . . . . . . . . . . . . . . . . . 18 𝑛(𝑥𝑋𝑗 / 𝑛𝐵)
171169, 170nfeq 2761 . . . . . . . . . . . . . . . . 17 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)
172168, 171nfim 1812 . . . . . . . . . . . . . . . 16 𝑛((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
173141anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑𝑗 ∈ (0...𝑀))))
174 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
175143mpteq2dv 4667 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝑗 / 𝑛𝐵))
176174, 175eqeq12d 2624 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵)))
177173, 176imbi12d 332 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵)) ↔ ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))))
178172, 177, 68chvar 2249 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
179166, 167, 178syl2anc 690 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗) = (𝑥𝑋𝑗 / 𝑛𝐵))
180179eqcomd 2615 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑥𝑋𝑗 / 𝑛𝐵) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗))
181180oveq2d 6543 . . . . . . . . . . . 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 23439 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ (𝑥𝑋𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
184124, 182, 128, 183syl3anc 1317 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)))
185184eqcomd 2615 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
186149adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
187166, 186jca 552 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
188 nfv 1829 . . . . . . . . . . . . . . 15 𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
189 nfcv 2750 . . . . . . . . . . . . . . . 16 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1))
19055, 156nfmpt 4668 . . . . . . . . . . . . . . . 16 𝑛(𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
191189, 190nfeq 2761 . . . . . . . . . . . . . . 15 𝑛((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)
192188, 191nfim 1812 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
193159anbi2d 735 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → ((𝜑𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
194 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)))
195161mpteq2dv 4667 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑗 + 1) → (𝑥𝑋𝐵) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
196194, 195eqeq12d 2624 . . . . . . . . . . . . . . 15 (𝑛 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵) ↔ ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
197193, 196imbi12d 332 . . . . . . . . . . . . . 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 3236 . . . . . . . . . . . . 13 ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵)))
199186, 187, 198sylc 62 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘(𝑗 + 1)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
200181, 185, 1993eqtrd 2647 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋𝑗 / 𝑛𝐵)) = (𝑥𝑋(𝑗 + 1) / 𝑛𝐵))
20135adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
20237adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
203131, 148, 165, 200, 201, 202dvmptdivc 23479 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥𝑋 ↦ (𝑗 / 𝑛𝐵 / 𝐶))) = (𝑥𝑋 ↦ ((𝑗 + 1) / 𝑛𝐵 / 𝐶)))
204203adantr 479 . . . . . . . . 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 2647 . . . . . . . 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 2615 . . . . . . 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 2648 . . . . . 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 2647 . . . . 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 1316 . . . 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 1255 . . 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 12406 . 2 (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶))))
2121, 2, 211sylc 62 1 ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  csb 3498  wss 3539  {cpr 4126  cmpt 4637  wf 5786  cfv 5790  (class class class)co 6527  pm cpm 7723  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796   / cdiv 10536  0cn0 11142  cuz 11522  ...cfz 12155  ..^cfzo 12292   D cdv 23378   D𝑛 cdvn 23379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872  ax-mulf 9873
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-supp 7161  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-ixp 7773  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fsupp 8137  df-fi 8178  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-icc 12012  df-fz 12156  df-fzo 12293  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-mulr 15731  df-starv 15732  df-sca 15733  df-vsca 15734  df-ip 15735  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-hom 15742  df-cco 15743  df-rest 15855  df-topn 15856  df-0g 15874  df-gsum 15875  df-topgen 15876  df-pt 15877  df-prds 15880  df-xrs 15934  df-qtop 15939  df-imas 15940  df-xps 15942  df-mre 16018  df-mrc 16019  df-acs 16021  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-mulg 17313  df-cntz 17522  df-cmn 17967  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-fbas 19513  df-fg 19514  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cld 20581  df-ntr 20582  df-cls 20583  df-nei 20660  df-lp 20698  df-perf 20699  df-cn 20789  df-cnp 20790  df-haus 20877  df-tx 21123  df-hmeo 21316  df-fil 21408  df-fm 21500  df-flim 21501  df-flf 21502  df-xms 21883  df-ms 21884  df-tms 21885  df-cncf 22437  df-limc 23381  df-dv 23382  df-dvn 23383
This theorem is referenced by: (None)
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