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Theorem dvnadd 25810

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 6884	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0))
2		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑀 + 𝑛) = (𝑀 + 0))
3	2	fveq2d 6888	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)))
4	1, 3	eqeq12d 2742	. . . . 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 6884	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
7		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑀 + 𝑛) = (𝑀 + 𝑘))
8	7	fveq2d 6888	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)))
9	6, 8	eqeq12d 2742	. . . . 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 6884	. . . . . 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 7412	. . . . . . 7 ⊢ (𝑛 = (𝑘 + 1) → (𝑀 + 𝑛) = (𝑀 + (𝑘 + 1)))
13	12	fveq2d 6888	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))
14	11, 13	eqeq12d 2742	. . . . 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 6884	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁))
17		oveq2 7412	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑀 + 𝑛) = (𝑀 + 𝑁))
18	17	fveq2d 6888	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))
19	16, 18	eqeq12d 2742	. . . . 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 25784	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
22	21	ad2antrr 723	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
23		ssidd 4000	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ℂ ⊆ ℂ)
24		cnex 11190	. . . . . . . . . . 11 ⊢ ℂ ∈ V
25		elpm2g 8837	. . . . . . . . . . 11 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
26	24, 25	mpan 687	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
27	26	simplbda 499	. . . . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → 𝑆 ∈ {ℝ, ℂ})
30		pmss12g 8862	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
32	31	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
33		dvnff 25804	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
34	33	ffvelcdmda 7079	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm dom 𝐹))
35	32, 34	sseldd 3978	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25805	. . . . . 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 583	. . . . 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 12483	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
39	38	adantl 481	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑀 ∈ ℂ)
40	39	addridd 11415	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (𝑀 + 0) = 𝑀)
41	40	fveq2d 6888	. . . . 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 2769	. . . 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 7412	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
45	35	adantr 480	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
46		simpr 484	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
47		dvnp1 25806	. . . . . . . . 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 1368	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ ℂ)
50		nn0cn 12483	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
51	50	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
52		1cnd 11210	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℂ)
53	49, 51, 52	addassd 11237	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 + 𝑘) + 1) = (𝑀 + (𝑘 + 1)))
54	53	fveq2d 6888	. . . . . . . . 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 773	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
56		nn0addcl 12508	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (𝑀 + 𝑘) ∈ ℕ₀)
57	56	adantll 711	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝑀 + 𝑘) ∈ ℕ₀)
58		dvnp1 25806	. . . . . . . . . 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 1368	. . . . . . . . 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 2768	. . . . . . . 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 2742	. . . . . . 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 413	. . . . 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 12658	. . 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 454	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 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 {cpr 4625 dom cdm 5669 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ↑_pm cpm 8820 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 ℕ₀cn0 12473 D cdv 25743 D^𝑛 cdvn 25744

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-icc 13334 df-fz 13488 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cnp 23083 df-haus 23170 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-limc 25746 df-dv 25747 df-dvn 25748

This theorem is referenced by: dvn2bss 25811 dvtaylp 26256 dvntaylp 26257 dvntaylp0 26258

W3C validator