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Theorem dvntaylp 25874

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	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvntaylp.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
dvntaylp.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
dvntaylp.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
dvntaylp.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
dvntaylp.b	⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)))

**Assertion**
Ref	Expression
dvntaylp	⊢ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))

Proof of Theorem dvntaylp Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvntaylp.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
2		nn0uz 12860	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
3	1, 2	eleqtrdi 2843	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
4		eluzfz2b 13506	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘0) ↔ 𝑀 ∈ (0...𝑀))
5	3, 4	sylib 217	. . 3 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
6		fveq2 6888	. . . . . 6 ⊢ (𝑚 = 0 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0))
7		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = 0 → ((𝑆 D^𝑛 𝐹)‘𝑚) = ((𝑆 D^𝑛 𝐹)‘0))
8	7	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0)))
9		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = 0 → (𝑀 − 𝑚) = (𝑀 − 0))
10	9	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 0 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 0)))
11		eqidd 2733	. . . . . . 7 ⊢ (𝑚 = 0 → 𝐵 = 𝐵)
12	8, 10, 11	oveq123d 7426	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵))
13	6, 12	eqeq12d 2748	. . . . 5 ⊢ (𝑚 = 0 → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵)))
14	13	imbi2d 340	. . . 4 ⊢ (𝑚 = 0 → ((𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵))))
15		fveq2 6888	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))
16		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘𝑚) = ((𝑆 D^𝑛 𝐹)‘𝑛))
17	16	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛)))
18		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑀 − 𝑚) = (𝑀 − 𝑛))
19	18	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 𝑛)))
20		eqidd 2733	. . . . . . 7 ⊢ (𝑚 = 𝑛 → 𝐵 = 𝐵)
21	17, 19, 20	oveq123d 7426	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵))
22	15, 21	eqeq12d 2748	. . . . 5 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)))
23	22	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵))))
24		fveq2 6888	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)))
25		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘𝑚) = ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))
26	25	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1))))
27		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (𝑀 − 𝑚) = (𝑀 − (𝑛 + 1)))
28	27	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − (𝑛 + 1))))
29		eqidd 2733	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → 𝐵 = 𝐵)
30	26, 28, 29	oveq123d 7426	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵))
31	24, 30	eqeq12d 2748	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵)))
32	31	imbi2d 340	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
33		fveq2 6888	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀))
34		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑆 D^𝑛 𝐹)‘𝑚) = ((𝑆 D^𝑛 𝐹)‘𝑀))
35	34	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀)))
36		oveq2 7413	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑀 − 𝑚) = (𝑀 − 𝑀))
37	36	oveq2d 7421	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 𝑀)))
38		eqidd 2733	. . . . . . 7 ⊢ (𝑚 = 𝑀 → 𝐵 = 𝐵)
39	35, 37, 38	oveq123d 7426	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
40	33, 39	eqeq12d 2748	. . . . 5 ⊢ (𝑚 = 𝑀 → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)))
41	40	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))))
42		ssidd 4004	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
43		mapsspm 8866	. . . . . . . 8 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
44		dvntaylp.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
45		dvntaylp.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
46		dvntaylp.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
47		dvntaylp.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
48	47, 1	nn0addcld 12532	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 𝑀) ∈ ℕ₀)
49		dvntaylp.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)))
50		eqid 2732	. . . . . . . . . 10 ⊢ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)
51	44, 45, 46, 48, 49, 50	taylpf 25869	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
52		cnex 11187	. . . . . . . . . 10 ⊢ ℂ ∈ V
53	52, 52	elmap 8861	. . . . . . . . 9 ⊢ (((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_m ℂ) ↔ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ)
54	51, 53	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_m ℂ))
55	43, 54	sselid 3979	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_pm ℂ))
56		dvn0 25432	. . . . . . 7 ⊢ ((ℂ ⊆ ℂ ∧ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
57	42, 55, 56	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
58		recnprss 25412	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
59	44, 58	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
60	52	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ∈ V)
61		elpm2r 8835	. . . . . . . . . 10 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
62	60, 44, 45, 46, 61	syl22anc 837	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
63		dvn0 25432	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
64	59, 62, 63	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
65	64	oveq2d 7421	. . . . . . 7 ⊢ (𝜑 → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0)) = (𝑆 Tayl 𝐹))
66	1	nn0cnd 12530	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67	66	subid1d 11556	. . . . . . . 8 ⊢ (𝜑 → (𝑀 − 0) = 𝑀)
68	67	oveq2d 7421	. . . . . . 7 ⊢ (𝜑 → (𝑁 + (𝑀 − 0)) = (𝑁 + 𝑀))
69		eqidd 2733	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝐵)
70	65, 68, 69	oveq123d 7426	. . . . . 6 ⊢ (𝜑 → ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))
71	57, 70	eqtr4d 2775	. . . . 5 ⊢ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵))
72	71	a1i 11	. . . 4 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘0))𝐵)))
73		oveq2 7413	. . . . . . 7 ⊢ (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵) → (ℂ D ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)))
74		ssidd 4004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ℂ ⊆ ℂ)
75	55	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_pm ℂ))
76		elfzouz 13632	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (ℤ_≥‘0))
77	76	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (ℤ_≥‘0))
78	77, 2	eleqtrrdi 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℕ₀)
79		dvnp1 25433	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)))
80	74, 75, 78, 79	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)))
81	44	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
82	62	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
83		dvnf 25435	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑛 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑛):dom ((𝑆 D^𝑛 𝐹)‘𝑛)⟶ℂ)
84	81, 82, 78, 83	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 𝐹)‘𝑛):dom ((𝑆 D^𝑛 𝐹)‘𝑛)⟶ℂ)
85		dvnbss 25436	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑛 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
86	81, 82, 78, 85	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
87	45	fdmd 6725	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = 𝐴)
88	87	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom 𝐹 = 𝐴)
89	86, 88	sseqtrd 4021	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D^𝑛 𝐹)‘𝑛) ⊆ 𝐴)
90	46	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐴 ⊆ 𝑆)
91	89, 90	sstrd 3991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D^𝑛 𝐹)‘𝑛) ⊆ 𝑆)
92	47	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
93		fzofzp1 13725	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0..^𝑀) → (𝑛 + 1) ∈ (0...𝑀))
94	93	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑛 + 1) ∈ (0...𝑀))
95		fznn0sub 13529	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (0...𝑀) → (𝑀 − (𝑛 + 1)) ∈ ℕ₀)
96	94, 95	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℕ₀)
97	92, 96	nn0addcld 12532	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − (𝑛 + 1))) ∈ ℕ₀)
98	49	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)))
99		elfzofz 13644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (0...𝑀))
100	99	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (0...𝑀))
101		fznn0sub 13529	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝑀) → (𝑀 − 𝑛) ∈ ℕ₀)
102	100, 101	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − 𝑛) ∈ ℕ₀)
103	92, 102	nn0addcld 12532	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − 𝑛)) ∈ ℕ₀)
104		dvnadd 25437	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ (𝑛 ∈ ℕ₀ ∧ (𝑁 + (𝑀 − 𝑛)) ∈ ℕ₀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛))) = ((𝑆 D^𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛)))))
105	81, 82, 78, 103, 104	syl22anc 837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛))) = ((𝑆 D^𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛)))))
106	47	nn0cnd 12530	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℂ)
107	106	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℂ)
108	96	nn0cnd 12530	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℂ)
109		1cnd 11205	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 1 ∈ ℂ)
110	107, 108, 109	addassd 11232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)))
111	66	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑀 ∈ ℂ)
112	78	nn0cnd 12530	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℂ)
113	111, 112, 109	nppcan2d 11593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑀 − (𝑛 + 1)) + 1) = (𝑀 − 𝑛))
114	113	oveq2d 7421	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)) = (𝑁 + (𝑀 − 𝑛)))
115	110, 114	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + (𝑀 − 𝑛)))
116	115	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛))))
117	112, 111	pncan3d 11570	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑛 + (𝑀 − 𝑛)) = 𝑀)
118	117	oveq2d 7421	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀 − 𝑛))) = (𝑁 + 𝑀))
119	111, 112	subcld 11567	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − 𝑛) ∈ ℂ)
120	107, 112, 119	add12d 11436	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀 − 𝑛))) = (𝑛 + (𝑁 + (𝑀 − 𝑛))))
121	118, 120	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + 𝑀) = (𝑛 + (𝑁 + (𝑀 − 𝑛))))
122	121	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛)))))
123	105, 116, 122	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)))
124	123	dmeqd 5903	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = dom ((𝑆 D^𝑛 𝐹)‘(𝑁 + 𝑀)))
125	98, 124	eleqtrrd 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)))
126	81, 84, 91, 97, 125	dvtaylp 25873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛)))𝐵))
127	115	oveq1d 7420	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵))
128	127	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)))
129	59	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
130		dvnp1 25433	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑛 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛)))
131	129, 82, 78, 130	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛)))
132	131	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1))) = (𝑆 Tayl (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛))))
133	132	eqcomd 2738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛))) = (𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1))))
134	133	oveqd 7422	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D^𝑛 𝐹)‘𝑛)))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵))
135	126, 128, 134	3eqtr3rd 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)))
136	80, 135	eqeq12d 2748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵) ↔ (ℂ D ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵))))
137	73, 136	imbitrrid 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵)))
138	137	expcom 414	. . . . 5 ⊢ (𝑛 ∈ (0..^𝑀) → (𝜑 → (((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
139	138	a2d 29	. . . 4 ⊢ (𝑛 ∈ (0..^𝑀) → ((𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑛))𝐵)) → (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑛 + 1)))𝐵))))
140	14, 23, 32, 41, 72, 139	fzind2 13746	. . 3 ⊢ (𝑀 ∈ (0...𝑀) → (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)))
141	5, 140	mpcom 38	. 2 ⊢ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
142	66	subidd 11555	. . . . 5 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
143	142	oveq2d 7421	. . . 4 ⊢ (𝜑 → (𝑁 + (𝑀 − 𝑀)) = (𝑁 + 0))
144	106	addridd 11410	. . . 4 ⊢ (𝜑 → (𝑁 + 0) = 𝑁)
145	143, 144	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝑁 + (𝑀 − 𝑀)) = 𝑁)
146	145	oveq1d 7420	. 2 ⊢ (𝜑 → ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵) = (𝑁(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
147	141, 146	eqtrd 2772	1 ⊢ (𝜑 → ((ℂ D^𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 {cpr 4629 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 ↑_pm cpm 8817 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11440 ℕ₀cn0 12468 ℤ_≥cuz 12818 ...cfz 13480 ..^cfzo 13623 D cdv 25371 D^𝑛 cdvn 25372 Tayl ctayl 25856

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tsms 23622 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-dvn 25376 df-tayl 25858

This theorem is referenced by: dvntaylp0 25875 taylthlem1 25876

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