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Theorem dvntaylp 25636
Description: The 𝑀-th derivative of the Taylor polynomial is the Taylor polynomial of the 𝑀-th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
Hypotheses
Ref Expression
dvntaylp.s (𝜑𝑆 ∈ {ℝ, ℂ})
dvntaylp.f (𝜑𝐹:𝐴⟶ℂ)
dvntaylp.a (𝜑𝐴𝑆)
dvntaylp.m (𝜑𝑀 ∈ ℕ0)
dvntaylp.n (𝜑𝑁 ∈ ℕ0)
dvntaylp.b (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
Assertion
Ref Expression
dvntaylp (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))

Proof of Theorem dvntaylp
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvntaylp.m . . . . 5 (𝜑𝑀 ∈ ℕ0)
2 nn0uz 12726 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2848 . . . 4 (𝜑𝑀 ∈ (ℤ‘0))
4 eluzfz2b 13371 . . . 4 (𝑀 ∈ (ℤ‘0) ↔ 𝑀 ∈ (0...𝑀))
53, 4sylib 217 . . 3 (𝜑𝑀 ∈ (0...𝑀))
6 fveq2 6830 . . . . . 6 (𝑚 = 0 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0))
7 fveq2 6830 . . . . . . . 8 (𝑚 = 0 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘0))
87oveq2d 7358 . . . . . . 7 (𝑚 = 0 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)))
9 oveq2 7350 . . . . . . . 8 (𝑚 = 0 → (𝑀𝑚) = (𝑀 − 0))
109oveq2d 7358 . . . . . . 7 (𝑚 = 0 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − 0)))
11 eqidd 2738 . . . . . . 7 (𝑚 = 0 → 𝐵 = 𝐵)
128, 10, 11oveq123d 7363 . . . . . 6 (𝑚 = 0 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))
136, 12eqeq12d 2753 . . . . 5 (𝑚 = 0 → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵)))
1413imbi2d 341 . . . 4 (𝑚 = 0 → ((𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))))
15 fveq2 6830 . . . . . 6 (𝑚 = 𝑛 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))
16 fveq2 6830 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑛))
1716oveq2d 7358 . . . . . . 7 (𝑚 = 𝑛 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛)))
18 oveq2 7350 . . . . . . . 8 (𝑚 = 𝑛 → (𝑀𝑚) = (𝑀𝑛))
1918oveq2d 7358 . . . . . . 7 (𝑚 = 𝑛 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑛)))
20 eqidd 2738 . . . . . . 7 (𝑚 = 𝑛𝐵 = 𝐵)
2117, 19, 20oveq123d 7363 . . . . . 6 (𝑚 = 𝑛 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))
2215, 21eqeq12d 2753 . . . . 5 (𝑚 = 𝑛 → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))
2322imbi2d 341 . . . 4 (𝑚 = 𝑛 → ((𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))))
24 fveq2 6830 . . . . . 6 (𝑚 = (𝑛 + 1) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)))
25 fveq2 6830 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))
2625oveq2d 7358 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
27 oveq2 7350 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (𝑀𝑚) = (𝑀 − (𝑛 + 1)))
2827oveq2d 7358 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − (𝑛 + 1))))
29 eqidd 2738 . . . . . . 7 (𝑚 = (𝑛 + 1) → 𝐵 = 𝐵)
3026, 28, 29oveq123d 7363 . . . . . 6 (𝑚 = (𝑛 + 1) → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))
3124, 30eqeq12d 2753 . . . . 5 (𝑚 = (𝑛 + 1) → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)))
3231imbi2d 341 . . . 4 (𝑚 = (𝑛 + 1) → ((𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
33 fveq2 6830 . . . . . 6 (𝑚 = 𝑀 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀))
34 fveq2 6830 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑀))
3534oveq2d 7358 . . . . . . 7 (𝑚 = 𝑀 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀)))
36 oveq2 7350 . . . . . . . 8 (𝑚 = 𝑀 → (𝑀𝑚) = (𝑀𝑀))
3736oveq2d 7358 . . . . . . 7 (𝑚 = 𝑀 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑀)))
38 eqidd 2738 . . . . . . 7 (𝑚 = 𝑀𝐵 = 𝐵)
3935, 37, 38oveq123d 7363 . . . . . 6 (𝑚 = 𝑀 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
4033, 39eqeq12d 2753 . . . . 5 (𝑚 = 𝑀 → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)))
4140imbi2d 341 . . . 4 (𝑚 = 𝑀 → ((𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))))
42 ssidd 3959 . . . . . . 7 (𝜑 → ℂ ⊆ ℂ)
43 mapsspm 8740 . . . . . . . 8 (ℂ ↑m ℂ) ⊆ (ℂ ↑pm ℂ)
44 dvntaylp.s . . . . . . . . . 10 (𝜑𝑆 ∈ {ℝ, ℂ})
45 dvntaylp.f . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ℂ)
46 dvntaylp.a . . . . . . . . . 10 (𝜑𝐴𝑆)
47 dvntaylp.n . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
4847, 1nn0addcld 12403 . . . . . . . . . 10 (𝜑 → (𝑁 + 𝑀) ∈ ℕ0)
49 dvntaylp.b . . . . . . . . . 10 (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
50 eqid 2737 . . . . . . . . . 10 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)
5144, 45, 46, 48, 49, 50taylpf 25631 . . . . . . . . 9 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
52 cnex 11058 . . . . . . . . . 10 ℂ ∈ V
5352, 52elmap 8735 . . . . . . . . 9 (((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m ℂ) ↔ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
5451, 53sylibr 233 . . . . . . . 8 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m ℂ))
5543, 54sselid 3934 . . . . . . 7 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
56 dvn0 25194 . . . . . . 7 ((ℂ ⊆ ℂ ∧ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
5742, 55, 56syl2anc 585 . . . . . 6 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
58 recnprss 25174 . . . . . . . . . 10 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
5944, 58syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
6052a1i 11 . . . . . . . . . 10 (𝜑 → ℂ ∈ V)
61 elpm2r 8709 . . . . . . . . . 10 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
6260, 44, 45, 46, 61syl22anc 837 . . . . . . . . 9 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
63 dvn0 25194 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6459, 62, 63syl2anc 585 . . . . . . . 8 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6564oveq2d 7358 . . . . . . 7 (𝜑 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)) = (𝑆 Tayl 𝐹))
661nn0cnd 12401 . . . . . . . . 9 (𝜑𝑀 ∈ ℂ)
6766subid1d 11427 . . . . . . . 8 (𝜑 → (𝑀 − 0) = 𝑀)
6867oveq2d 7358 . . . . . . 7 (𝜑 → (𝑁 + (𝑀 − 0)) = (𝑁 + 𝑀))
69 eqidd 2738 . . . . . . 7 (𝜑𝐵 = 𝐵)
7065, 68, 69oveq123d 7363 . . . . . 6 (𝜑 → ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
7157, 70eqtr4d 2780 . . . . 5 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))
7271a1i 11 . . . 4 (𝑀 ∈ (ℤ‘0) → (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵)))
73 oveq2 7350 . . . . . . 7 (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → (ℂ D ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))
74 ssidd 3959 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
7555adantr 482 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
76 elfzouz 13497 . . . . . . . . . . 11 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (ℤ‘0))
7776adantl 483 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (ℤ‘0))
7877, 2eleqtrrdi 2849 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℕ0)
79 dvnp1 25195 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)))
8074, 75, 78, 79syl3anc 1371 . . . . . . . 8 ((𝜑𝑛 ∈ (0..^𝑀)) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)))
8144adantr 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
8262adantr 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
83 dvnf 25197 . . . . . . . . . . 11 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
8481, 82, 78, 83syl3anc 1371 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
85 dvnbss 25198 . . . . . . . . . . . . 13 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
8681, 82, 78, 85syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
8745fdmd 6667 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = 𝐴)
8887adantr 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom 𝐹 = 𝐴)
8986, 88sseqtrd 3976 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝐴)
9046adantr 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐴𝑆)
9189, 90sstrd 3946 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝑆)
9247adantr 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℕ0)
93 fzofzp1 13590 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^𝑀) → (𝑛 + 1) ∈ (0...𝑀))
9493adantl 483 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + 1) ∈ (0...𝑀))
95 fznn0sub 13394 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (0...𝑀) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9694, 95syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9792, 96nn0addcld 12403 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − (𝑛 + 1))) ∈ ℕ0)
9849adantr 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
99 elfzofz 13509 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (0...𝑀))
10099adantl 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (0...𝑀))
101 fznn0sub 13394 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...𝑀) → (𝑀𝑛) ∈ ℕ0)
102100, 101syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℕ0)
10392, 102nn0addcld 12403 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀𝑛)) ∈ ℕ0)
104 dvnadd 25199 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑛 ∈ ℕ0 ∧ (𝑁 + (𝑀𝑛)) ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
10581, 82, 78, 103, 104syl22anc 837 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
10647nn0cnd 12401 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℂ)
107106adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℂ)
10896nn0cnd 12401 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℂ)
109 1cnd 11076 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 1 ∈ ℂ)
110107, 108, 109addassd 11103 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)))
11166adantr 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑀 ∈ ℂ)
11278nn0cnd 12401 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℂ)
113111, 112, 109nppcan2d 11464 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑀 − (𝑛 + 1)) + 1) = (𝑀𝑛))
114113oveq2d 7358 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)) = (𝑁 + (𝑀𝑛)))
115110, 114eqtrd 2777 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + (𝑀𝑛)))
116115fveq2d 6834 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))))
117112, 111pncan3d 11441 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + (𝑀𝑛)) = 𝑀)
118117oveq2d 7358 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑁 + 𝑀))
119111, 112subcld 11438 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℂ)
120107, 112, 119add12d 11307 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑛 + (𝑁 + (𝑀𝑛))))
121118, 120eqtr3d 2779 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + 𝑀) = (𝑛 + (𝑁 + (𝑀𝑛))))
122121fveq2d 6834 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
123105, 116, 1223eqtr4d 2787 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
124123dmeqd 5852 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
12598, 124eleqtrrd 2841 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)))
12681, 84, 91, 97, 125dvtaylp 25635 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))𝐵))
127115oveq1d 7357 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))
128127oveq2d 7358 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) = (ℂ D ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))
12959adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
130 dvnp1 25195 . . . . . . . . . . . . 13 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
131129, 82, 78, 130syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
132131oveq2d 7358 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))) = (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))))
133132eqcomd 2743 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
134133oveqd 7359 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))
135126, 128, 1343eqtr3rd 2786 . . . . . . . 8 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵) = (ℂ D ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))
13680, 135eqeq12d 2753 . . . . . . 7 ((𝜑𝑛 ∈ (0..^𝑀)) → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵) ↔ (ℂ D ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))))
13773, 136syl5ibr 246 . . . . . 6 ((𝜑𝑛 ∈ (0..^𝑀)) → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)))
138137expcom 415 . . . . 5 (𝑛 ∈ (0..^𝑀) → (𝜑 → (((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
139138a2d 29 . . . 4 (𝑛 ∈ (0..^𝑀) → ((𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) → (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
14014, 23, 32, 41, 72, 139fzind2 13611 . . 3 (𝑀 ∈ (0...𝑀) → (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)))
1415, 140mpcom 38 . 2 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
14266subidd 11426 . . . . 5 (𝜑 → (𝑀𝑀) = 0)
143142oveq2d 7358 . . . 4 (𝜑 → (𝑁 + (𝑀𝑀)) = (𝑁 + 0))
144106addid1d 11281 . . . 4 (𝜑 → (𝑁 + 0) = 𝑁)
145143, 144eqtrd 2777 . . 3 (𝜑 → (𝑁 + (𝑀𝑀)) = 𝑁)
146145oveq1d 7357 . 2 (𝜑 → ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
147141, 146eqtrd 2777 1 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1541  wcel 2106  Vcvv 3442  wss 3902  {cpr 4580  dom cdm 5625  wf 6480  cfv 6484  (class class class)co 7342  m cmap 8691  pm cpm 8692  cc 10975  cr 10976  0cc0 10977  1c1 10978   + caddc 10980  cmin 11311  0cn0 12339  cuz 12688  ...cfz 13345  ..^cfzo 13488   D cdv 25133   D𝑛 cdvn 25134   Tayl ctayl 25618
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 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-inf2 9503  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054  ax-pre-sup 11055  ax-addf 11056  ax-mulf 11057
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-om 7786  df-1st 7904  df-2nd 7905  df-supp 8053  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-2o 8373  df-er 8574  df-map 8693  df-pm 8694  df-ixp 8762  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-fsupp 9232  df-fi 9273  df-sup 9304  df-inf 9305  df-oi 9372  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-div 11739  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-dec 12544  df-uz 12689  df-q 12795  df-rp 12837  df-xneg 12954  df-xadd 12955  df-xmul 12956  df-icc 13192  df-fz 13346  df-fzo 13489  df-seq 13828  df-exp 13889  df-fac 14094  df-hash 14151  df-cj 14910  df-re 14911  df-im 14912  df-sqrt 15046  df-abs 15047  df-clim 15297  df-sum 15498  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-starv 17075  df-sca 17076  df-vsca 17077  df-ip 17078  df-tset 17079  df-ple 17080  df-ds 17082  df-unif 17083  df-hom 17084  df-cco 17085  df-rest 17231  df-topn 17232  df-0g 17250  df-gsum 17251  df-topgen 17252  df-pt 17253  df-prds 17256  df-xrs 17311  df-qtop 17316  df-imas 17317  df-xps 17319  df-mre 17393  df-mrc 17394  df-acs 17396  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-submnd 18529  df-grp 18677  df-minusg 18678  df-mulg 18798  df-cntz 19020  df-cmn 19484  df-abl 19485  df-mgp 19816  df-ur 19833  df-ring 19880  df-cring 19881  df-psmet 20695  df-xmet 20696  df-met 20697  df-bl 20698  df-mopn 20699  df-fbas 20700  df-fg 20701  df-cnfld 20704  df-top 22149  df-topon 22166  df-topsp 22188  df-bases 22202  df-cld 22276  df-ntr 22277  df-cls 22278  df-nei 22355  df-lp 22393  df-perf 22394  df-cn 22484  df-cnp 22485  df-haus 22572  df-tx 22819  df-hmeo 23012  df-fil 23103  df-fm 23195  df-flim 23196  df-flf 23197  df-tsms 23384  df-xms 23579  df-ms 23580  df-tms 23581  df-cncf 24147  df-limc 25136  df-dv 25137  df-dvn 25138  df-tayl 25620
This theorem is referenced by:  dvntaylp0  25637  taylthlem1  25638
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