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Theorem dvntaylp 26428
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 12918 . . . . 5 0 = (ℤ‘0)
31, 2eleqtrdi 2849 . . . 4 (𝜑𝑀 ∈ (ℤ‘0))
4 eluzfz2b 13570 . . . 4 (𝑀 ∈ (ℤ‘0) ↔ 𝑀 ∈ (0...𝑀))
53, 4sylib 218 . . 3 (𝜑𝑀 ∈ (0...𝑀))
6 fveq2 6907 . . . . . 6 (𝑚 = 0 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0))
7 fveq2 6907 . . . . . . . 8 (𝑚 = 0 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘0))
87oveq2d 7447 . . . . . . 7 (𝑚 = 0 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)))
9 oveq2 7439 . . . . . . . 8 (𝑚 = 0 → (𝑀𝑚) = (𝑀 − 0))
109oveq2d 7447 . . . . . . 7 (𝑚 = 0 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − 0)))
11 eqidd 2736 . . . . . . 7 (𝑚 = 0 → 𝐵 = 𝐵)
128, 10, 11oveq123d 7452 . . . . . 6 (𝑚 = 0 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))
136, 12eqeq12d 2751 . . . . 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 340 . . . 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 6907 . . . . . 6 (𝑚 = 𝑛 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))
16 fveq2 6907 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑛))
1716oveq2d 7447 . . . . . . 7 (𝑚 = 𝑛 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛)))
18 oveq2 7439 . . . . . . . 8 (𝑚 = 𝑛 → (𝑀𝑚) = (𝑀𝑛))
1918oveq2d 7447 . . . . . . 7 (𝑚 = 𝑛 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑛)))
20 eqidd 2736 . . . . . . 7 (𝑚 = 𝑛𝐵 = 𝐵)
2117, 19, 20oveq123d 7452 . . . . . 6 (𝑚 = 𝑛 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))
2215, 21eqeq12d 2751 . . . . 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 340 . . . 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 6907 . . . . . 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 6907 . . . . . . . 8 (𝑚 = (𝑛 + 1) → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))
2625oveq2d 7447 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
27 oveq2 7439 . . . . . . . 8 (𝑚 = (𝑛 + 1) → (𝑀𝑚) = (𝑀 − (𝑛 + 1)))
2827oveq2d 7447 . . . . . . 7 (𝑚 = (𝑛 + 1) → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀 − (𝑛 + 1))))
29 eqidd 2736 . . . . . . 7 (𝑚 = (𝑛 + 1) → 𝐵 = 𝐵)
3026, 28, 29oveq123d 7452 . . . . . 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 2751 . . . . 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 340 . . . 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 6907 . . . . . 6 (𝑚 = 𝑀 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀))
34 fveq2 6907 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑀))
3534oveq2d 7447 . . . . . . 7 (𝑚 = 𝑀 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀)))
36 oveq2 7439 . . . . . . . 8 (𝑚 = 𝑀 → (𝑀𝑚) = (𝑀𝑀))
3736oveq2d 7447 . . . . . . 7 (𝑚 = 𝑀 → (𝑁 + (𝑀𝑚)) = (𝑁 + (𝑀𝑀)))
38 eqidd 2736 . . . . . . 7 (𝑚 = 𝑀𝐵 = 𝐵)
3935, 37, 38oveq123d 7452 . . . . . 6 (𝑚 = 𝑀 → ((𝑁 + (𝑀𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
4033, 39eqeq12d 2751 . . . . 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 340 . . . 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 4019 . . . . . . 7 (𝜑 → ℂ ⊆ ℂ)
43 mapsspm 8915 . . . . . . . 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 12589 . . . . . . . . . 10 (𝜑 → (𝑁 + 𝑀) ∈ ℕ0)
49 dvntaylp.b . . . . . . . . . 10 (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
50 eqid 2735 . . . . . . . . . 10 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)
5144, 45, 46, 48, 49, 50taylpf 26422 . . . . . . . . 9 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
52 cnex 11234 . . . . . . . . . 10 ℂ ∈ V
5352, 52elmap 8910 . . . . . . . . 9 (((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m ℂ) ↔ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
5451, 53sylibr 234 . . . . . . . 8 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m ℂ))
5543, 54sselid 3993 . . . . . . 7 (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
56 dvn0 25975 . . . . . . 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 584 . . . . . 6 (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
58 recnprss 25954 . . . . . . . . . 10 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
5944, 58syl 17 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
6052a1i 11 . . . . . . . . . 10 (𝜑 → ℂ ∈ V)
61 elpm2r 8884 . . . . . . . . . 10 (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
6260, 44, 45, 46, 61syl22anc 839 . . . . . . . . 9 (𝜑𝐹 ∈ (ℂ ↑pm 𝑆))
63 dvn0 25975 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6459, 62, 63syl2anc 584 . . . . . . . 8 (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)
6564oveq2d 7447 . . . . . . 7 (𝜑 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)) = (𝑆 Tayl 𝐹))
661nn0cnd 12587 . . . . . . . . 9 (𝜑𝑀 ∈ ℂ)
6766subid1d 11607 . . . . . . . 8 (𝜑 → (𝑀 − 0) = 𝑀)
6867oveq2d 7447 . . . . . . 7 (𝜑 → (𝑁 + (𝑀 − 0)) = (𝑁 + 𝑀))
69 eqidd 2736 . . . . . . 7 (𝜑𝐵 = 𝐵)
7065, 68, 69oveq123d 7452 . . . . . 6 (𝜑 → ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
7157, 70eqtr4d 2778 . . . . 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 7439 . . . . . . 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 4019 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
7555adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm ℂ))
76 elfzouz 13700 . . . . . . . . . . 11 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (ℤ‘0))
7776adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (ℤ‘0))
7877, 2eleqtrrdi 2850 . . . . . . . . 9 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℕ0)
79 dvnp1 25976 . . . . . . . . 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 1370 . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
8262adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐹 ∈ (ℂ ↑pm 𝑆))
83 dvnf 25978 . . . . . . . . . . 11 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
8481, 82, 78, 83syl3anc 1370 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ)
85 dvnbss 25979 . . . . . . . . . . . . 13 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
8681, 82, 78, 85syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
8745fdmd 6747 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = 𝐴)
8887adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → dom 𝐹 = 𝐴)
8986, 88sseqtrd 4036 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝐴)
9046adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐴𝑆)
9189, 90sstrd 4006 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝑆)
9247adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℕ0)
93 fzofzp1 13800 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^𝑀) → (𝑛 + 1) ∈ (0...𝑀))
9493adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + 1) ∈ (0...𝑀))
95 fznn0sub 13593 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ (0...𝑀) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9694, 95syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℕ0)
9792, 96nn0addcld 12589 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − (𝑛 + 1))) ∈ ℕ0)
9849adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
99 elfzofz 13712 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (0...𝑀))
10099adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (0...𝑀))
101 fznn0sub 13593 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...𝑀) → (𝑀𝑛) ∈ ℕ0)
102100, 101syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℕ0)
10392, 102nn0addcld 12589 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀𝑛)) ∈ ℕ0)
104 dvnadd 25980 . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
10647nn0cnd 12587 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℂ)
107106adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℂ)
10896nn0cnd 12587 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℂ)
109 1cnd 11254 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → 1 ∈ ℂ)
110107, 108, 109addassd 11281 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)))
11166adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑀 ∈ ℂ)
11278nn0cnd 12587 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℂ)
113111, 112, 109nppcan2d 11644 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑀 − (𝑛 + 1)) + 1) = (𝑀𝑛))
114113oveq2d 7447 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)) = (𝑁 + (𝑀𝑛)))
115110, 114eqtrd 2775 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + (𝑀𝑛)))
116115fveq2d 6911 . . . . . . . . . . . . 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 11621 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑛 + (𝑀𝑛)) = 𝑀)
118117oveq2d 7447 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑁 + 𝑀))
119111, 112subcld 11618 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑀𝑛) ∈ ℂ)
120107, 112, 119add12d 11486 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀𝑛))) = (𝑛 + (𝑁 + (𝑀𝑛))))
121118, 120eqtr3d 2777 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑁 + 𝑀) = (𝑛 + (𝑁 + (𝑀𝑛))))
122121fveq2d 6911 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀𝑛)))))
123105, 116, 1223eqtr4d 2785 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
124123dmeqd 5919 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))
12598, 124eleqtrrd 2842 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)))
12681, 84, 91, 97, 125dvtaylp 26427 . . . . . . . . 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 7446 . . . . . . . . . 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 7447 . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
130 dvnp1 25976 . . . . . . . . . . . . 13 ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
131129, 82, 78, 130syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))
132131oveq2d 7447 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))) = (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))))
133132eqcomd 2741 . . . . . . . . . 10 ((𝜑𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))))
134133oveqd 7448 . . . . . . . . 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 2784 . . . . . . . 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 2751 . . . . . . 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, 136imbitrrid 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 413 . . . . 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 13821 . . 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 11606 . . . . 5 (𝜑 → (𝑀𝑀) = 0)
143142oveq2d 7447 . . . 4 (𝜑 → (𝑁 + (𝑀𝑀)) = (𝑁 + 0))
144106addridd 11459 . . . 4 (𝜑 → (𝑁 + 0) = 𝑁)
145143, 144eqtrd 2775 . . 3 (𝜑 → (𝑁 + (𝑀𝑀)) = 𝑁)
146145oveq1d 7446 . 2 (𝜑 → ((𝑁 + (𝑀𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))
147141, 146eqtrd 2775 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 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  {cpr 4633  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  pm cpm 8866  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  cmin 11490  0cn0 12524  cuz 12876  ...cfz 13544  ..^cfzo 13691   D cdv 25913   D𝑛 cdvn 25914   Tayl ctayl 26409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-fac 14310  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-ur 20200  df-ring 20253  df-cring 20254  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-tsms 24151  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-dvn 25918  df-tayl 26411
This theorem is referenced by:  dvntaylp0  26429  taylthlem1  26430
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