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Theorem dvnadd 25437

Description: The 𝑁-th derivative of the 𝑀-th derivative of 𝐹 is the same as the 𝑀 + 𝑁-th derivative of 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
dvnadd	⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))

Proof of Theorem dvnadd Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0))
2		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑀 + 𝑛) = (𝑀 + 0))
3	2	fveq2d 6892	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)))
4	1, 3	eqeq12d 2748	. . . . 5 ⊢ (𝑛 = 0 → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) ↔ ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0))))
5	4	imbi2d 340	. . . 4 ⊢ (𝑛 = 0 → ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛))) ↔ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)))))
6		fveq2 6888	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
7		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑀 + 𝑛) = (𝑀 + 𝑘))
8	7	fveq2d 6892	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)))
9	6, 8	eqeq12d 2748	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) ↔ ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))))
10	9	imbi2d 340	. . . 4 ⊢ (𝑛 = 𝑘 → ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛))) ↔ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)))))
11		fveq2 6888	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)))
12		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = (𝑘 + 1) → (𝑀 + 𝑛) = (𝑀 + (𝑘 + 1)))
13	12	fveq2d 6892	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))
14	11, 13	eqeq12d 2748	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) ↔ ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1)))))
15	14	imbi2d 340	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛))) ↔ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))))
16		fveq2 6888	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁))
17		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑀 + 𝑛) = (𝑀 + 𝑁))
18	17	fveq2d 6892	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))
19	16, 18	eqeq12d 2748	. . . . 5 ⊢ (𝑛 = 𝑁 → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) ↔ ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁))))
20	19	imbi2d 340	. . . 4 ⊢ (𝑛 = 𝑁 → ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛))) ↔ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))))
21		recnprss 25412	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
22	21	ad2antrr 724	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
23		ssidd 4004	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ℂ ⊆ ℂ)
24		cnex 11187	. . . . . . . . . . 11 ⊢ ℂ ∈ V
25		elpm2g 8834	. . . . . . . . . . 11 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
26	24, 25	mpan 688	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
27	26	simplbda 500	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → dom 𝐹 ⊆ 𝑆)
28	24	a1i 11	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ℂ ∈ V)
29		simpl 483	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → 𝑆 ∈ {ℝ, ℂ})
30		pmss12g 8859	. . . . . . . . 9 ⊢ (((ℂ ⊆ ℂ ∧ dom 𝐹 ⊆ 𝑆) ∧ (ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ})) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
31	23, 27, 28, 29, 30	syl22anc 837	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
32	31	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
33		dvnff 25431	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
34	33	ffvelcdmda 7083	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm dom 𝐹))
35	32, 34	sseldd 3982	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25432	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑀))
37	22, 35, 36	syl2anc 584	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑀))
38		nn0cn 12478	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
39	38	adantl 482	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑀 ∈ ℂ)
40	39	addridd 11410	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (𝑀 + 0) = 𝑀)
41	40	fveq2d 6892	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)) = ((𝑆 D^𝑛 𝐹)‘𝑀))
42	37, 41	eqtr4d 2775	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)))
43		oveq2 7413	. . . . . . 7 ⊢ (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)) → (𝑆 D ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))))
44	22	adantr 481	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
45	35	adantr 481	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
46		simpr 485	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
47		dvnp1 25433	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ℂ ∧ ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = (𝑆 D ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘)))
48	44, 45, 46, 47	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = (𝑆 D ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘)))
49	39	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ ℂ)
50		nn0cn 12478	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
51	50	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
52		1cnd 11205	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℂ)
53	49, 51, 52	addassd 11232	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 + 𝑘) + 1) = (𝑀 + (𝑘 + 1)))
54	53	fveq2d 6892	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘((𝑀 + 𝑘) + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))
55		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
56		nn0addcl 12503	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑀 + 𝑘) ∈ ℕ₀)
57	56	adantll 712	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑀 + 𝑘) ∈ ℕ₀)
58		dvnp1 25433	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ (𝑀 + 𝑘) ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘((𝑀 + 𝑘) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))))
59	44, 55, 57, 58	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘((𝑀 + 𝑘) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))))
60	54, 59	eqtr3d 2774	. . . . . . . 8 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))))
61	48, 60	eqeq12d 2748	. . . . . . 7 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))) ↔ (𝑆 D ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)))))
62	43, 61	imbitrrid 245	. . . . . 6 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1)))))
63	62	expcom 414	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))))
64	63	a2d 29	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘))) → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑘 + 1)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))))
65	5, 10, 15, 20, 42, 64	nn0ind 12653	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁))))
66	65	com12 32	. 2 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ ℕ₀ → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁))))
67	66	impr 455	1 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 {cpr 4629 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_pm cpm 8817 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ℕ₀cn0 12468 D cdv 25371 D^𝑛 cdvn 25372

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

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-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-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 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-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 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-cnp 22723 df-haus 22810 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-limc 25374 df-dv 25375 df-dvn 25376

This theorem is referenced by: dvn2bss 25438 dvtaylp 25873 dvntaylp 25874 dvntaylp0 25875

W3C validator