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Theorem binomfallfac 16007

Description: A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)

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

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑁

Proof of Theorem binomfallfac Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7395	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 0))
2		oveq2 7395	. . . . . . . 8 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3		fz0sn 13588	. . . . . . . 8 ⊢ (0...0) = {0}
4	2, 3	eqtrdi 2780	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = {0})
5		oveq1 7394	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
6		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑚 − 𝑘) = (0 − 𝑘))
7	6	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (0 − 𝑘)))
8	7	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))
9	5, 8	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = 0 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝑚 = 0 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
11	4, 10	sumeq12dv 15672	. . . . . 6 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
12	1, 11	eqeq12d 2745	. . . . 5 ⊢ (𝑚 = 0 → (((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) ↔ ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))))
13	12	imbi2d 340	. . . 4 ⊢ (𝑚 = 0 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))))
14		oveq2 7395	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑛))
15		oveq2 7395	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
16		oveq1 7394	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
17		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 𝑘) = (𝑛 − 𝑘))
18	17	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑛 − 𝑘)))
19	18	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))
20	16, 19	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
22	15, 21	sumeq12dv 15672	. . . . . 6 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
23	14, 22	eqeq12d 2745	. . . . 5 ⊢ (𝑚 = 𝑛 → (((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) ↔ ((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))))
24	23	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑛 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))))
25		oveq2 7395	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac (𝑛 + 1)))
26		oveq2 7395	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
27		oveq1 7394	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
28		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 𝑘) = ((𝑛 + 1) − 𝑘))
29	28	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac ((𝑛 + 1) − 𝑘)))
30	29	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))
31	27, 30	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
32	31	adantr 480	. . . . . . 7 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
33	26, 32	sumeq12dv 15672	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
34	25, 33	eqeq12d 2745	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) ↔ ((𝐴 + 𝐵) FallFac (𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))))
35	34	imbi2d 340	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac (𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))))
36		oveq2 7395	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑁))
37		oveq2 7395	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
38		oveq1 7394	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
39		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑘) = (𝑁 − 𝑘))
40	39	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑁 − 𝑘)))
41	40	oveq1d 7402	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))
42	38, 41	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
43	42	adantr 480	. . . . . . 7 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
44	37, 43	sumeq12dv 15672	. . . . . 6 ⊢ (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
45	36, 44	eqeq12d 2745	. . . . 5 ⊢ (𝑚 = 𝑁 → (((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) ↔ ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))))
46	45	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑚) = Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))))
47		fallfac0 15994	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
48		fallfac0 15994	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 FallFac 0) = 1)
49	47, 48	oveqan12d 7406	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = (1 · 1))
50		1t1e1 12343	. . . . . . . 8 ⊢ (1 · 1) = 1
51	49, 50	eqtrdi 2780	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = 1)
52	51	oveq2d 7403	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = (1 · 1))
53	52, 50	eqtrdi 2780	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = 1)
54		0cn 11166	. . . . . 6 ⊢ 0 ∈ ℂ
55		ax-1cn 11126	. . . . . . 7 ⊢ 1 ∈ ℂ
56	53, 55	eqeltrdi 2836	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ)
57		oveq2 7395	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
58		0nn0 12457	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
59		bcnn 14277	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
60	58, 59	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
61	57, 60	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
62		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
63		0m0e0 12301	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
64	62, 63	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
65	64	oveq2d 7403	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴 FallFac (0 − 𝑘)) = (𝐴 FallFac 0))
66		oveq2 7395	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵 FallFac 𝑘) = (𝐵 FallFac 0))
67	65, 66	oveq12d 7405	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac 0) · (𝐵 FallFac 0)))
68	61, 67	oveq12d 7405	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
69	68	sumsn 15712	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ) → Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
70	54, 56, 69	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
71		addcl 11150	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
72		fallfac0 15994	. . . . . 6 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((𝐴 + 𝐵) FallFac 0) = 1)
73	71, 72	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = 1)
74	53, 70, 73	3eqtr4rd 2775	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
75		simprl 770	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
76		simprr 772	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
77		simpl 482	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ₀)
78		id 22	. . . . . . 7 ⊢ (((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))) → ((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
79	75, 76, 77, 78	binomfallfaclem2 16006	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) ∧ ((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))) → ((𝐴 + 𝐵) FallFac (𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
80	79	exp31 419	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))) → ((𝐴 + 𝐵) FallFac (𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))))
81	80	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac (𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))))
82	13, 24, 35, 46, 74, 81	nn0ind 12629	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))))
83	82	com12 32	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑁 ∈ ℕ₀ → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))))
84	83	3impia 1117	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {csn 4589 (class class class)co 7387 ℂcc 11066 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 ℕ₀cn0 12442 ...cfz 13468 Ccbc 14267 Σcsu 15652 FallFac cfallfac 15970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-prod 15870 df-risefac 15972 df-fallfac 15973

This theorem is referenced by: binomrisefac 16008

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