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Theorem binomrisefac 15977

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 11450	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
2	1	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
3	2	oveq1d 7383	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 + 𝐵) FallFac 𝑁) = ((-𝐴 + -𝐵) FallFac 𝑁))
4		negcl 11392	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
5		negcl 11392	. . . . . 6 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
6		id 22	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
7		binomfallfac 15976	. . . . . 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 1161	. . . . 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 2772	. . . 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 7384	. . 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 13908	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
12		neg1cn 12142	. . . . . 6 ⊢ -1 ∈ ℂ
13		expcl 14014	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
14	12, 13	mpan 691	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (-1↑𝑁) ∈ ℂ)
15	14	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
16		simp3 1139	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
17		elfzelz 13452	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
18		bccl 14257	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
19	16, 17, 18	syl2an 597	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ₀)
20	19	nn0cnd 12476	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ)
21		simpl1 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
22	21	negcld 11491	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -𝐴 ∈ ℂ)
23	16	nn0zd 12525	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
24		zsubcl 12545	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ)
25	23, 17, 24	syl2an 597	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ)
26		elfzle2 13456	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ≤ 𝑁)
27	26	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≤ 𝑁)
28		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
29	28	nn0red 12475	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
30		elfznn0 13548	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
31	30	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
32	31	nn0red 12475	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
33	29, 32	subge0d 11739	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (0 ≤ (𝑁 − 𝑘) ↔ 𝑘 ≤ 𝑁))
34	27, 33	mpbird 257	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 0 ≤ (𝑁 − 𝑘))
35		elnn0z 12513	. . . . . . . 8 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ ↔ ((𝑁 − 𝑘) ∈ ℤ ∧ 0 ≤ (𝑁 − 𝑘)))
36	25, 34, 35	sylanbrc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℕ₀)
37		fallfaccl 15951	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
38	22, 36, 37	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
39		simp2 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐵 ∈ ℂ)
40	39	negcld 11491	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -𝐵 ∈ ℂ)
41		fallfaccl 15951	. . . . . . 7 ⊢ ((-𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-𝐵 FallFac 𝑘) ∈ ℂ)
42	40, 30, 41	syl2an 597	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐵 FallFac 𝑘) ∈ ℂ)
43	38, 42	mulcld 11164	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)) ∈ ℂ)
44	20, 43	mulcld 11164	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))) ∈ ℂ)
45	11, 15, 44	fsummulc2 15719	. . 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 2772	. 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 11120	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
48		risefallfac 15959	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
49	47, 48	stoic3 1778	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
50		risefallfac 15959	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
51	21, 36, 50	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
52		simpl2 1194	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ)
53		risefallfac 15959	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
54	52, 31, 53	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
55	51, 54	oveq12d 7386	. . . . . 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 14014	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
57	12, 36, 56	sylancr 588	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
58		expcl 14014	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
59	12, 30, 58	sylancr 588	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → (-1↑𝑘) ∈ ℂ)
60	59	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑘) ∈ ℂ)
61	57, 38, 60, 42	mul4d 11357	. . . . . 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 14083	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)))
64	16	nn0cnd 12476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
65	30	nn0cnd 12476	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ)
66		npcan 11401	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
67	64, 65, 66	syl2an 597	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
68	67	oveq2d 7384	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = (-1↑𝑁))
69	63, 68	eqtr3d 2774	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑁))
70	69	oveq1d 7383	. . . . . 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 2776	. . . . 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 7384	. . . 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 480	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑁) ∈ ℂ)
74	20, 73, 43	mul12d 11354	. . . 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 2772	. . 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 15637	. 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 2782	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 (class class class)co 7368 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ≤ cle 11179 − cmin 11376 -cneg 11377 ℕ₀cn0 12413 ℤcz 12500 ...cfz 13435 ↑cexp 13996 Ccbc 14237 Σcsu 15621 FallFac cfallfac 15939 RiseFac crisefac 15940

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-prod 15839 df-risefac 15941 df-fallfac 15942

This theorem is referenced by: (None)

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