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Theorem dvnadd 25877

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 6900	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘0))
2		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑀 + 𝑛) = (𝑀 + 0))
3	2	fveq2d 6904	. . . . . 6 ⊢ (𝑛 = 0 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 0)))
4	1, 3	eqeq12d 2743	. . . . 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 339	. . . 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 6900	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
7		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑀 + 𝑛) = (𝑀 + 𝑘))
8	7	fveq2d 6904	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑘)))
9	6, 8	eqeq12d 2743	. . . . 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 339	. . . 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 6900	. . . . . 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 7432	. . . . . . 7 ⊢ (𝑛 = (𝑘 + 1) → (𝑀 + 𝑛) = (𝑀 + (𝑘 + 1)))
13	12	fveq2d 6904	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑘 + 1))))
14	11, 13	eqeq12d 2743	. . . . 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 339	. . . 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 6900	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑛) = ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑁))
17		oveq2 7432	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑀 + 𝑛) = (𝑀 + 𝑁))
18	17	fveq2d 6904	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑛)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + 𝑁)))
19	16, 18	eqeq12d 2743	. . . . 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 339	. . . 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 25851	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
22	21	ad2antrr 724	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
23		ssidd 4003	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ℂ ⊆ ℂ)
24		cnex 11225	. . . . . . . . . . 11 ⊢ ℂ ∈ V
25		elpm2g 8867	. . . . . . . . . . 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 498	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → 𝑆 ∈ {ℝ, ℂ})
30		pmss12g 8892	. . . . . . . . 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 479	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (ℂ ↑_pm dom 𝐹) ⊆ (ℂ ↑_pm 𝑆))
33		dvnff 25871	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → (𝑆 D^𝑛 𝐹):ℕ₀⟶(ℂ ↑_pm dom 𝐹))
34	33	ffvelcdmda 7097	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm dom 𝐹))
35	32, 34	sseldd 3981	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25872	. . . . . 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 582	. . . . 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 12518	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
39	38	adantl 480	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → 𝑀 ∈ ℂ)
40	39	addridd 11450	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) → (𝑀 + 0) = 𝑀)
41	40	fveq2d 6904	. . . . 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 2770	. . . 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 7432	. . . . . . 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 479	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑆 ⊆ ℂ)
45	35	adantr 479	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
46		simpr 483	. . . . . . . . 9 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
47		dvnp1 25873	. . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ ℂ)
50		nn0cn 12518	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℂ)
51	50	adantl 480	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℂ)
52		1cnd 11245	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 1 ∈ ℂ)
53	49, 51, 52	addassd 11272	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ 𝑀 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑀 + 𝑘) + 1) = (𝑀 + (𝑘 + 1)))
54	53	fveq2d 6904	. . . . . . . . 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 12543	. . . . . . . . . . 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 25873	. . . . . . . . . 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 2769	. . . . . . . 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 2743	. . . . . . 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 412	. . . . 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 12693	. . 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 453	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 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 {cpr 4632 dom cdm 5680 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ↑_pm cpm 8850 ℂcc 11142 ℝcr 11143 0cc0 11144 1c1 11145 + caddc 11147 ℕ₀cn0 12508 D cdv 25810 D^𝑛 cdvn 25811

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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fi 9440 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-icc 13369 df-fz 13523 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-mulr 17252 df-starv 17253 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-rest 17409 df-topn 17410 df-topgen 17430 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cnp 23150 df-haus 23237 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-limc 25813 df-dv 25814 df-dvn 25815

This theorem is referenced by: dvn2bss 25878 dvtaylp 26323 dvntaylp 26324 dvntaylp0 26325

W3C validator