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Theorem binomrisefac 16091

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 11510	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
2	1	3adant3 1150	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
3	2	oveq1d 7425	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 + 𝐵) FallFac 𝑁) = ((-𝐴 + -𝐵) FallFac 𝑁))
4		negcl 11452	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
5		negcl 11452	. . . . . 6 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
6		id 23	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
7		binomfallfac 16090	. . . . . 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 1178	. . . . 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 2798	. . . 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 7426	. . 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 14005	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
12		neg1cn 12198	. . . . . 6 ⊢ -1 ∈ ℂ
13		expcl 14111	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
14	12, 13	mpan 702	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (-1↑𝑁) ∈ ℂ)
15	14	3ad2ant3 1153	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
16		simp3 1156	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
17		elfzelz 13547	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
18		bccl 14354	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
19	16, 17, 18	syl2an 607	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ₀)
20	19	nn0cnd 12562	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ)
21		simpl1 1210	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
22	21	negcld 11551	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -𝐴 ∈ ℂ)
23	16	nn0zd 12611	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
24		zsubcl 12631	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ)
25	23, 17, 24	syl2an 607	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ)
26		elfzle2 13551	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ≤ 𝑁)
27	26	adantl 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≤ 𝑁)
28		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
29	28	nn0red 12561	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
30		elfznn0 13644	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
31	30	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
32	31	nn0red 12561	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
33	29, 32	subge0d 11799	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (0 ≤ (𝑁 − 𝑘) ↔ 𝑘 ≤ 𝑁))
34	27, 33	mpbird 260	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 0 ≤ (𝑁 − 𝑘))
35		elnn0z 12599	. . . . . . . 8 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ ↔ ((𝑁 − 𝑘) ∈ ℤ ∧ 0 ≤ (𝑁 − 𝑘)))
36	25, 34, 35	sylanbrc 594	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℕ₀)
37		fallfaccl 16066	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
38	22, 36, 37	syl2anc 595	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
39		simp2 1155	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐵 ∈ ℂ)
40	39	negcld 11551	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -𝐵 ∈ ℂ)
41		fallfaccl 16066	. . . . . . 7 ⊢ ((-𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-𝐵 FallFac 𝑘) ∈ ℂ)
42	40, 30, 41	syl2an 607	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-𝐵 FallFac 𝑘) ∈ ℂ)
43	38, 42	mulcld 11224	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)) ∈ ℂ)
44	20, 43	mulcld 11224	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))) ∈ ℂ)
45	11, 15, 44	fsummulc2 15831	. . 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 2798	. 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 11177	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
48		risefallfac 16074	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
49	47, 48	stoic3 1806	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
50		risefallfac 16074	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
51	21, 36, 50	syl2anc 595	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 RiseFac (𝑁 − 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-𝐴 FallFac (𝑁 − 𝑘))))
52		simpl2 1211	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ)
53		risefallfac 16074	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
54	52, 31, 53	syl2anc 595	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐵 RiseFac 𝑘) = ((-1↑𝑘) · (-𝐵 FallFac 𝑘)))
55	51, 54	oveq12d 7428	. . . . . 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 14111	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
57	12, 36, 56	sylancr 598	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑(𝑁 − 𝑘)) ∈ ℂ)
58		expcl 14111	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (-1↑𝑘) ∈ ℂ)
59	12, 30, 58	sylancr 598	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → (-1↑𝑘) ∈ ℂ)
60	59	adantl 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑘) ∈ ℂ)
61	57, 38, 60, 42	mul4d 11417	. . . . . 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 14180	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)))
64	16	nn0cnd 12562	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
65	30	nn0cnd 12562	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ)
66		npcan 11461	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
67	64, 65, 66	syl2an 607	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
68	67	oveq2d 7426	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = (-1↑𝑁))
69	63, 68	eqtr3d 2800	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑁))
70	69	oveq1d 7425	. . . . . 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 2802	. . . . 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 7426	. . . 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 485	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑁) ∈ ℂ)
74	20, 73, 43	mul12d 11414	. . . 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 2798	. . 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 15749	. 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 2808	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 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 class class class wbr 5109 (class class class)co 7410 ℂcc 11093 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 ≤ cle 11239 − cmin 11436 -cneg 11437 ℕ₀cn0 12499 ℤcz 12586 ...cfz 13530 ↑cexp 14093 Ccbc 14334 Σcsu 15733 FallFac cfallfac 16054 RiseFac crisefac 16055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-prod 15954 df-risefac 16056 df-fallfac 16057

This theorem is referenced by: (None)

W3C validator