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Theorem binomfallfac 15950

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 7360	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 0))
2		oveq2 7360	. . . . . . . 8 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3		fz0sn 13529	. . . . . . . 8 ⊢ (0...0) = {0}
4	2, 3	eqtrdi 2784	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = {0})
5		oveq1 7359	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
6		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑚 − 𝑘) = (0 − 𝑘))
7	6	oveq2d 7368	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (0 − 𝑘)))
8	7	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))
9	5, 8	oveq12d 7370	. . . . . . . 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 15615	. . . . . 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 2749	. . . . 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 7360	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑛))
15		oveq2 7360	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
16		oveq1 7359	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
17		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 𝑘) = (𝑛 − 𝑘))
18	17	oveq2d 7368	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑛 − 𝑘)))
19	18	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))
20	16, 19	oveq12d 7370	. . . . . . . 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 15615	. . . . . 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 2749	. . . . 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 7360	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac (𝑛 + 1)))
26		oveq2 7360	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
27		oveq1 7359	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
28		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 𝑘) = ((𝑛 + 1) − 𝑘))
29	28	oveq2d 7368	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac ((𝑛 + 1) − 𝑘)))
30	29	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))
31	27, 30	oveq12d 7370	. . . . . . . 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 15615	. . . . . 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 2749	. . . . 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 7360	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑁))
37		oveq2 7360	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
38		oveq1 7359	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
39		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑘) = (𝑁 − 𝑘))
40	39	oveq2d 7368	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑁 − 𝑘)))
41	40	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))
42	38, 41	oveq12d 7370	. . . . . . . 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 15615	. . . . . 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 2749	. . . . 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 15937	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
48		fallfac0 15937	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 FallFac 0) = 1)
49	47, 48	oveqan12d 7371	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = (1 · 1))
50		1t1e1 12289	. . . . . . . 8 ⊢ (1 · 1) = 1
51	49, 50	eqtrdi 2784	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = 1)
52	51	oveq2d 7368	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = (1 · 1))
53	52, 50	eqtrdi 2784	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = 1)
54		0cn 11111	. . . . . 6 ⊢ 0 ∈ ℂ
55		ax-1cn 11071	. . . . . . 7 ⊢ 1 ∈ ℂ
56	53, 55	eqeltrdi 2841	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ)
57		oveq2 7360	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
58		0nn0 12403	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
59		bcnn 14221	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
60	58, 59	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
61	57, 60	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
62		oveq2 7360	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
63		0m0e0 12247	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
64	62, 63	eqtrdi 2784	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
65	64	oveq2d 7368	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴 FallFac (0 − 𝑘)) = (𝐴 FallFac 0))
66		oveq2 7360	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵 FallFac 𝑘) = (𝐵 FallFac 0))
67	65, 66	oveq12d 7370	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac 0) · (𝐵 FallFac 0)))
68	61, 67	oveq12d 7370	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
69	68	sumsn 15655	. . . . . 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 11095	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
72		fallfac0 15937	. . . . . 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 2779	. . . 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 15949	. . . . . 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 12574	. . 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 1541 ∈ wcel 2113 {csn 4575 (class class class)co 7352 ℂcc 11011 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 − cmin 11351 ℕ₀cn0 12388 ...cfz 13409 Ccbc 14211 Σcsu 15595 FallFac cfallfac 15913

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-fac 14183 df-bc 14212 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-sum 15596 df-prod 15813 df-risefac 15915 df-fallfac 15916

This theorem is referenced by: binomrisefac 15951

W3C validator