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Theorem binomrisefac 15752

Description: A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)

**Assertion**
Ref	Expression
binomrisefac	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑁

**Proof of Theorem binomrisefac**
Step	Hyp	Ref	Expression
1		negdi 11278	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
2	1	3adant3 1131	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
3	2	oveq1d 7290	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 + 𝐵) FallFac 𝑁) = ((-𝐴 + -𝐵) FallFac 𝑁))
4		negcl 11221	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
5		negcl 11221	. . . . . 6 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
6		id 22	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
7		binomfallfac 15751	. . . . . 6 ⊢ ((-𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((-𝐴 + -𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
8	4, 5, 6, 7	syl3an 1159	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((-𝐴 + -𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
9	3, 8	eqtrd 2778	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
10	9	oveq2d 7291	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)) = ((-1↑𝑁) · Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
11		fzfid 13693	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
12		neg1cn 12087	. . . . . 6 ⊢ -1 ∈ ℂ
13		expcl 13800	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
14	12, 13	mpan 687	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (-1↑𝑁) ∈ ℂ)
15	14	3ad2ant3 1134	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
16		simp3 1137	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
17		elfzelz 13256	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
18		bccl 14036	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
19	16, 17, 18	syl2an 596	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ₀)
20	19	nn0cnd 12295	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ)
21		simpl1 1190	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
22	21	negcld 11319	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -𝐴 ∈ ℂ)
23	16	nn0zd 12424	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
24		zsubcl 12362	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ)
25	23, 17, 24	syl2an 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ)
26		elfzle2 13260	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ≤ 𝑁)
27	26	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≤ 𝑁)
28		simpl3 1192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
29	28	nn0red 12294	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
30		elfznn0 13349	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
31	30	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
32	31	nn0red 12294	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
33	29, 32	subge0d 11565	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (0 ≤ (𝑁 − 𝑘) ↔ 𝑘 ≤ 𝑁))
34	27, 33	mpbird 256	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 0 ≤ (𝑁 − 𝑘))
35		elnn0z 12332	. . . . . . . 8 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ ↔ ((𝑁 − 𝑘) ∈ ℤ ∧ 0 ≤ (𝑁 − 𝑘)))
36	25, 34, 35	sylanbrc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℕ₀)
37		fallfaccl 15726	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
38	22, 36, 37	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
39		simp2 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐵 ∈ ℂ)
40	39	negcld 11319	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -𝐵 ∈ ℂ)
41		fallfaccl 15726	. . . . . . 7 ⊢ ((-𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-𝐵 FallFac 𝑘) ∈ ℂ)
42	40, 30, 41	syl2an 596	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐵 FallFac 𝑘) ∈ ℂ)
43	38, 42	mulcld 10995	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)) ∈ ℂ)
44	20, 43	mulcld 10995	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))) ∈ ℂ)
45	11, 15, 44	fsummulc2 15496	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((-1↑𝑁) · Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑁) · ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
46	10, 45	eqtrd 2778	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑁) · ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
47		addcl 10953	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
48		risefallfac 15734	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
49	47, 48	stoic3 1779	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
50		risefallfac 15734	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
51	21, 36, 50	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
52		simpl2 1191	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ)
53		risefallfac 15734	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
54	52, 31, 53	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
55	51, 54	oveq12d 7293	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘)) = (((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))) · ((-1↑𝑘) · (-𝐵 FallFac 𝑘))))
56		expcl 13800	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
57	12, 36, 56	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
58		expcl 13800	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
59	12, 30, 58	sylancr 587	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → (-1↑𝑘) ∈ ℂ)
60	59	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑘) ∈ ℂ)
61	57, 38, 60, 42	mul4d 11187	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))) · ((-1↑𝑘) · (-𝐵 FallFac 𝑘))) = (((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
62	12	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -1 ∈ ℂ)
63	62, 31, 36	expaddd 13866	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)))
64	16	nn0cnd 12295	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
65	30	nn0cnd 12295	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ)
66		npcan 11230	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
67	64, 65, 66	syl2an 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
68	67	oveq2d 7291	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = (-1↑𝑁))
69	63, 68	eqtr3d 2780	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑁))
70	69	oveq1d 7290	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))) = ((-1↑𝑁) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
71	55, 61, 70	3eqtrd 2782	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘)) = ((-1↑𝑁) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))))
72	71	oveq2d 7291	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = ((𝑁C𝑘) · ((-1↑𝑁) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
73	15	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑁) ∈ ℂ)
74	20, 73, 43	mul12d 11184	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((-1↑𝑁) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))) = ((-1↑𝑁) · ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
75	72, 74	eqtrd 2778	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = ((-1↑𝑁) · ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
76	75	sumeq2dv 15415	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑁) · ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)))))
77	46, 49, 76	3eqtr4d 2788	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 ℕ₀cn0 12233 ℤcz 12319 ...cfz 13239 ↑cexp 13782 Ccbc 14016 Σcsu 15397 FallFac cfallfac 15714 RiseFac crisefac 15715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-prod 15616 df-risefac 15716 df-fallfac 15717

This theorem is referenced by: (None)

W3C validator