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Theorem binomfallfac 16012

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 7421	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 0))
2		oveq2 7421	. . . . . . . 8 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3		fz0sn 13628	. . . . . . . 8 ⊢ (0...0) = {0}
4	2, 3	eqtrdi 2781	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = {0})
5		oveq1 7420	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
6		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑚 − 𝑘) = (0 − 𝑘))
7	6	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (0 − 𝑘)))
8	7	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))
9	5, 8	oveq12d 7431	. . . . . . . 8 ⊢ (𝑚 = 0 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
10	9	adantr 479	. . . . . . 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 15679	. . . . . 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 2741	. . . . 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 339	. . . 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 7421	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑛))
15		oveq2 7421	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
16		oveq1 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
17		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 𝑘) = (𝑛 − 𝑘))
18	17	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑛 − 𝑘)))
19	18	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))
20	16, 19	oveq12d 7431	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
21	20	adantr 479	. . . . . . 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 15679	. . . . . 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 2741	. . . . 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 339	. . . 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 7421	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac (𝑛 + 1)))
26		oveq2 7421	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
27		oveq1 7420	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
28		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 𝑘) = ((𝑛 + 1) − 𝑘))
29	28	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac ((𝑛 + 1) − 𝑘)))
30	29	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))
31	27, 30	oveq12d 7431	. . . . . . . 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 479	. . . . . . 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 15679	. . . . . 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 2741	. . . . 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 339	. . . 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 7421	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑁))
37		oveq2 7421	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
38		oveq1 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
39		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑘) = (𝑁 − 𝑘))
40	39	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑁 − 𝑘)))
41	40	oveq1d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))
42	38, 41	oveq12d 7431	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
43	42	adantr 479	. . . . . . 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 15679	. . . . . 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 2741	. . . . 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 339	. . . 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 15999	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
48		fallfac0 15999	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 FallFac 0) = 1)
49	47, 48	oveqan12d 7432	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = (1 · 1))
50		1t1e1 12399	. . . . . . . 8 ⊢ (1 · 1) = 1
51	49, 50	eqtrdi 2781	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = 1)
52	51	oveq2d 7429	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = (1 · 1))
53	52, 50	eqtrdi 2781	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = 1)
54		0cn 11231	. . . . . 6 ⊢ 0 ∈ ℂ
55		ax-1cn 11191	. . . . . . 7 ⊢ 1 ∈ ℂ
56	53, 55	eqeltrdi 2833	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ)
57		oveq2 7421	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
58		0nn0 12512	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
59		bcnn 14298	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
60	58, 59	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
61	57, 60	eqtrdi 2781	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
62		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
63		0m0e0 12357	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
64	62, 63	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
65	64	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴 FallFac (0 − 𝑘)) = (𝐴 FallFac 0))
66		oveq2 7421	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵 FallFac 𝑘) = (𝐵 FallFac 0))
67	65, 66	oveq12d 7431	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac 0) · (𝐵 FallFac 0)))
68	61, 67	oveq12d 7431	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
69	68	sumsn 15719	. . . . . 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 585	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
71		addcl 11215	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
72		fallfac0 15999	. . . . . 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 2776	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
75		simprl 769	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
76		simprr 771	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
77		simpl 481	. . . . . . 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 16011	. . . . . 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 418	. . . . 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 12682	. . 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 1114	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4625 (class class class)co 7413 ℂcc 11131 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 − cmin 11469 ℕ₀cn0 12497 ...cfz 13511 Ccbc 14288 Σcsu 15659 FallFac cfallfac 15975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-prod 15877 df-risefac 15977 df-fallfac 15978

This theorem is referenced by: binomrisefac 16013

W3C validator