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Theorem dvntaylp 24170
 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 11760 . . . . 5 0 = (ℤ‘0)
31, 2syl6eleq 2740 . . . 4 (𝜑𝑀 ∈ (ℤ‘0))
4 eluzfz2b 12388 . . . 4 (𝑀 ∈ (ℤ‘0) ↔ 𝑀 ∈ (0...𝑀))
53, 4sylib 208 . . 3 (𝜑𝑀 ∈ (0...𝑀))
6 fveq2 6229 . . . . . 6 (𝑚 = 0 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0))
7 fveq2 6229 . . . . . . . 8 (𝑚 = 0 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘0))
87oveq2d 6706 . . . . . . 7 (𝑚 = 0 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)))
9 oveq2 6698 . . . . . . . 8 (𝑚 = 0 → (𝑀𝑚) = (𝑀 − 0))
109oveq2d 6706 . . . . . . 7 (𝑚 = 0 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − 0)))
11 eqidd 2652 . . . . . . 7 (𝑚 = 0 → 𝐵 = 𝐵)
128, 10, 11oveq123d 6711 . . . . . 6 (𝑚 = 0 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))
136, 12eqeq12d 2666 . . . . 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 329 . . . 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 6229 . . . . . 6 (𝑚 = 𝑛 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))
16 fveq2 6229 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑛))
1716oveq2d 6706 . . . . . . 7 (𝑚 = 𝑛 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛)))
18 oveq2 6698 . . . . . . . 8 (𝑚 = 𝑛 → (𝑀𝑚) = (𝑀𝑛))
1918oveq2d 6706 . . . . . . 7 (𝑚 = 𝑛 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑛)))
20 eqidd 2652 . . . . . . 7 (𝑚 = 𝑛𝐵 = 𝐵)
2117, 19, 20oveq123d 6711 . . . . . 6 (𝑚 = 𝑛 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))
2215, 21eqeq12d 2666 . . . . 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 329 . . . 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 6229 . . . . . 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 6229 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))
2625oveq2d 6706 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
27 oveq2 6698 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (𝑀𝑚) = (𝑀 − (𝑛 + 1)))
2827oveq2d 6706 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − (𝑛 + 1))))
29 eqidd 2652 . . . . . . 7 (𝑚 = (𝑛 + 1) → 𝐵 = 𝐵)
3026, 28, 29oveq123d 6711 . . . . . 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 2666 . . . . 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 329 . . . 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 6229 . . . . . 6 (𝑚 = 𝑀 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀))
34 fveq2 6229 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑀))
3534oveq2d 6706 . . . . . . 7 (𝑚 = 𝑀 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀)))
36 oveq2 6698 . . . . . . . 8 (𝑚 = 𝑀 → (𝑀𝑚) = (𝑀𝑀))
3736oveq2d 6706 . . . . . . 7 (𝑚 = 𝑀 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑀)))
38 eqidd 2652 . . . . . . 7 (𝑚 = 𝑀𝐵 = 𝐵)
3935, 37, 38oveq123d 6711 . . . . . 6 (𝑚 = 𝑀 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
4033, 39eqeq12d 2666 . . . . 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 329 . . . 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 ssid 3657 . . . . . . . 8 ℂ ⊆ ℂ
4342a1i 11 . . . . . . 7 (𝜑 → ℂ ⊆ ℂ)
44 mapsspm 7933 . . . . . . . 8 (ℂ ↑𝑚 ℂ) ⊆ (ℂ ↑pm ℂ)
45 dvntaylp.s . . . . . . . . . 10 (𝜑𝑆 ∈ {ℝ, ℂ})
46 dvntaylp.f . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ℂ)
47 dvntaylp.a . . . . . . . . . 10 (𝜑𝐴𝑆)
48 dvntaylp.n . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
4948, 1nn0addcld 11393 . . . . . . . . . 10 (𝜑 → (𝑁 + 𝑀) ∈ ℕ0)
50 dvntaylp.b . . . . . . . . . 10 (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
51 eqid 2651 . . . . . . . . . 10 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)
5245, 46, 47, 49, 50, 51taylpf 24165 . . . . . . . . 9 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
53 cnex 10055 . . . . . . . . . 10 ℂ ∈ V
5453, 53elmap 7928 . . . . . . . . 9 (((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑𝑚 ℂ) ↔ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
5552, 54sylibr 224 . . . . . . . 8 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑𝑚 ℂ))
5644, 55sseldi 3634 . . . . . . 7 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
57 dvn0 23732 . . . . . . 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))
5843, 56, 57syl2anc 694 . . . . . 6 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
59 recnprss 23713 . . . . . . . . . 10 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
6045, 59syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
6153a1i 11 . . . . . . . . . 10 (𝜑 → ℂ ∈ V)
62 elpm2r 7917 . . . . . . . . . 10 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
6361, 45, 46, 47, 62syl22anc 1367 . . . . . . . . 9 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
64 dvn0 23732 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6560, 63, 64syl2anc 694 . . . . . . . 8 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6665oveq2d 6706 . . . . . . 7 (𝜑 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)) = (𝑆 Tayl 𝐹))
671nn0cnd 11391 . . . . . . . . 9 (𝜑𝑀 ∈ ℂ)
6867subid1d 10419 . . . . . . . 8 (𝜑 → (𝑀 − 0) = 𝑀)
6968oveq2d 6706 . . . . . . 7 (𝜑 → (𝑁 + (𝑀 − 0)) = (𝑁 + 𝑀))
70 eqidd 2652 . . . . . . 7 (𝜑𝐵 = 𝐵)
7166, 69, 70oveq123d 6711 . . . . . 6 (𝜑 → ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
7258, 71eqtr4d 2688 . . . . 5 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))
7372a1i 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)))
74 oveq2 6698 . . . . . . 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)))
7542a1i 11 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
7656adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
77 elfzouz 12513 . . . . . . . . . . 11 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (ℤ‘0))
7877adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (ℤ‘0))
7978, 2syl6eleqr 2741 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℕ0)
80 dvnp1 23733 . . . . . . . . 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)))
8175, 76, 79, 80syl3anc 1366 . . . . . . . 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)))
8245adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
8363adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
84 dvnf 23735 . . . . . . . . . . 11 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
8582, 83, 79, 84syl3anc 1366 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
86 dvnbss 23736 . . . . . . . . . . . . 13 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
8782, 83, 79, 86syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
88 fdm 6089 . . . . . . . . . . . . . 14 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
8946, 88syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = 𝐴)
9089adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom 𝐹 = 𝐴)
9187, 90sseqtrd 3674 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝐴)
9247adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐴𝑆)
9391, 92sstrd 3646 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝑆)
9448adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℕ0)
95 fzofzp1 12605 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^𝑀) → (𝑛 + 1) ∈ (0...𝑀))
9695adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + 1) ∈ (0...𝑀))
97 fznn0sub 12411 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (0...𝑀) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9896, 97syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9994, 98nn0addcld 11393 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − (𝑛 + 1))) ∈ ℕ0)
10050adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
101 elfzofz 12524 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (0...𝑀))
102101adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (0...𝑀))
103 fznn0sub 12411 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...𝑀) → (𝑀𝑛) ∈ ℕ0)
104102, 103syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℕ0)
10594, 104nn0addcld 11393 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀𝑛)) ∈ ℕ0)
106 dvnadd 23737 . . . . . . . . . . . . . 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)))))
10782, 83, 79, 105, 106syl22anc 1367 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
10848nn0cnd 11391 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℂ)
109108adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℂ)
11098nn0cnd 11391 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℂ)
111 1cnd 10094 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 1 ∈ ℂ)
112109, 110, 111addassd 10100 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)))
11367adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑀 ∈ ℂ)
11479nn0cnd 11391 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℂ)
115113, 114, 111nppcan2d 10456 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑀 − (𝑛 + 1)) + 1) = (𝑀𝑛))
116115oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)) = (𝑁 + (𝑀𝑛)))
117112, 116eqtrd 2685 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + (𝑀𝑛)))
118117fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))))
119114, 113pncan3d 10433 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + (𝑀𝑛)) = 𝑀)
120119oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑁 + 𝑀))
121113, 114subcld 10430 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℂ)
122109, 114, 121add12d 10300 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑛 + (𝑁 + (𝑀𝑛))))
123120, 122eqtr3d 2687 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + 𝑀) = (𝑛 + (𝑁 + (𝑀𝑛))))
124123fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
125107, 118, 1243eqtr4d 2695 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
126125dmeqd 5358 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
127100, 126eleqtrrd 2733 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)))
12882, 85, 93, 99, 127dvtaylp 24169 . . . . . . . . 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))
129117oveq1d 6705 . . . . . . . . . 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))
130129oveq2d 6706 . . . . . . . . 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)))
13160adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
132 dvnp1 23733 . . . . . . . . . . . . 13 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
133131, 83, 79, 132syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
134133oveq2d 6706 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))) = (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))))
135134eqcomd 2657 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
136135oveqd 6707 . . . . . . . . 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))
137128, 130, 1363eqtr3rd 2694 . . . . . . . 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)))
13881, 137eqeq12d 2666 . . . . . . 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))))
13974, 138syl5ibr 236 . . . . . 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)))
140139expcom 450 . . . . 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))))
141140a2d 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))))
14214, 23, 32, 41, 73, 141fzind2 12626 . . 3 (𝑀 ∈ (0...𝑀) → (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)))
1435, 142mpcom 38 . 2 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
14467subidd 10418 . . . . 5 (𝜑 → (𝑀𝑀) = 0)
145144oveq2d 6706 . . . 4 (𝜑 → (𝑁 + (𝑀𝑀)) = (𝑁 + 0))
146108addid1d 10274 . . . 4 (𝜑 → (𝑁 + 0) = 𝑁)
147145, 146eqtrd 2685 . . 3 (𝜑 → (𝑁 + (𝑀𝑀)) = 𝑁)
148147oveq1d 6705 . 2 (𝜑 → ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
149143, 148eqtrd 2685 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 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  {cpr 4212  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899   ↑pm cpm 7900  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   − cmin 10304  ℕ0cn0 11330  ℤ≥cuz 11725  ...cfz 12364  ..^cfzo 12504   D cdv 23672   D𝑛 cdvn 23673   Tayl ctayl 24152 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-fal 1529  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-fac 13101  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  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-grp 17472  df-minusg 17473  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  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-tsms 21977  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-dvn 23677  df-tayl 24154 This theorem is referenced by:  dvntaylp0  24171  taylthlem1  24172
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