binomrisefac - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > binomrisefac		Structured version Visualization version GIF version

Theorem binomrisefac 15824

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 11351	. . . . . . 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 7330	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (-(𝐴 + 𝐵) FallFac 𝑁) = ((-𝐴 + -𝐵) FallFac 𝑁))
4		negcl 11294	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
5		negcl 11294	. . . . . 6 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
6		id 22	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
7		binomfallfac 15823	. . . . . 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 2777	. . . 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 7331	. . 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 13766	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
12		neg1cn 12160	. . . . . 6 ⊢ -1 ∈ ℂ
13		expcl 13873	. . . . . 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 13329	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
18		bccl 14109	. . . . . . 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 12368	. . . . 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 11392	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -𝐴 ∈ ℂ)
23	16	nn0zd 12497	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
24		zsubcl 12435	. . . . . . . . 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 13333	. . . . . . . . . 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 12367	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
30		elfznn0 13422	. . . . . . . . . . . 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 12367	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
33	29, 32	subge0d 11638	. . . . . . . . 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 12405	. . . . . . . 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 15798	. . . . . . 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 11392	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -𝐵 ∈ ℂ)
41		fallfaccl 15798	. . . . . . 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 11068	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘)) ∈ ℂ)
44	20, 43	mulcld 11068	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((-𝐴 FallFac (𝑁 − 𝑘)) · (-𝐵 FallFac 𝑘))) ∈ ℂ)
45	11, 15, 44	fsummulc2 15568	. . 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 2777	. 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 11026	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
48		risefallfac 15806	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
49	47, 48	stoic3 1777	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = ((-1↑𝑁) · (-(𝐴 + 𝐵) FallFac 𝑁)))
50		risefallfac 15806	. . . . . . . 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 15806	. . . . . . . 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 7333	. . . . . 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 13873	. . . . . . . 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 13873	. . . . . . . . 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 11260	. . . . . 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 13939	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)))
64	16	nn0cnd 12368	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
65	30	nn0cnd 12368	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ)
66		npcan 11303	. . . . . . . . . 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 7331	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (-1↑((𝑁 − 𝑘) + 𝑘)) = (-1↑𝑁))
69	63, 68	eqtr3d 2779	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((-1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑁))
70	69	oveq1d 7330	. . . . . 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 2781	. . . . 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 7331	. . . 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 11257	. . . 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 2777	. . 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 15487	. 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 2787	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 1540 ∈ wcel 2105 class class class wbr 5087 (class class class)co 7315 ℂcc 10942 0cc0 10944 1c1 10945 + caddc 10947 · cmul 10949 ≤ cle 11083 − cmin 11278 -cneg 11279 ℕ₀cn0 12306 ℤcz 12392 ...cfz 13312 ↑cexp 13855 Ccbc 14089 Σcsu 15469 FallFac cfallfac 15786 RiseFac crisefac 15787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7628 ax-inf2 9470 ax-cnex 11000 ax-resscn 11001 ax-1cn 11002 ax-icn 11003 ax-addcl 11004 ax-addrcl 11005 ax-mulcl 11006 ax-mulrcl 11007 ax-mulcom 11008 ax-addass 11009 ax-mulass 11010 ax-distr 11011 ax-i2m1 11012 ax-1ne0 11013 ax-1rid 11014 ax-rnegex 11015 ax-rrecex 11016 ax-cnre 11017 ax-pre-lttri 11018 ax-pre-lttrn 11019 ax-pre-ltadd 11020 ax-pre-mulgt0 11021 ax-pre-sup 11022

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7272 df-ov 7318 df-oprab 7319 df-mpo 7320 df-om 7758 df-1st 7876 df-2nd 7877 df-frecs 8144 df-wrecs 8175 df-recs 8249 df-rdg 8288 df-1o 8344 df-er 8546 df-en 8782 df-dom 8783 df-sdom 8784 df-fin 8785 df-sup 9271 df-oi 9339 df-card 9768 df-pnf 11084 df-mnf 11085 df-xr 11086 df-ltxr 11087 df-le 11088 df-sub 11280 df-neg 11281 df-div 11706 df-nn 12047 df-2 12109 df-3 12110 df-n0 12307 df-z 12393 df-uz 12656 df-rp 12804 df-fz 13313 df-fzo 13456 df-seq 13795 df-exp 13856 df-fac 14061 df-bc 14090 df-hash 14118 df-cj 14882 df-re 14883 df-im 14884 df-sqrt 15018 df-abs 15019 df-clim 15269 df-sum 15470 df-prod 15688 df-risefac 15788 df-fallfac 15789

This theorem is referenced by: (None)

W3C validator