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Theorem binomfallfac 16120

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 7431	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 0))
2		oveq2 7431	. . . . . . . 8 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3		fz0sn 13674	. . . . . . . 8 ⊢ (0...0) = {0}
4	2, 3	eqtrdi 2817	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = {0})
5		oveq1 7430	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
6		oveq1 7430	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑚 − 𝑘) = (0 − 𝑘))
7	6	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (0 − 𝑘)))
8	7	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))
9	5, 8	oveq12d 7441	. . . . . . . 8 ⊢ (𝑚 = 0 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
10	9	adantr 486	. . . . . . 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 15783	. . . . . 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 2782	. . . . 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 343	. . . 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 7431	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑛))
15		oveq2 7431	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
16		oveq1 7430	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
17		oveq1 7430	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 𝑘) = (𝑛 − 𝑘))
18	17	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑛 − 𝑘)))
19	18	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))
20	16, 19	oveq12d 7441	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
21	20	adantr 486	. . . . . . 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 15783	. . . . . 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 2782	. . . . 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 343	. . . 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 7431	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac (𝑛 + 1)))
26		oveq2 7431	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
27		oveq1 7430	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
28		oveq1 7430	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 𝑘) = ((𝑛 + 1) − 𝑘))
29	28	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac ((𝑛 + 1) − 𝑘)))
30	29	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))
31	27, 30	oveq12d 7441	. . . . . . . 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 486	. . . . . . 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 15783	. . . . . 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 2782	. . . . 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 343	. . . 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 7431	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑁))
37		oveq2 7431	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
38		oveq1 7430	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
39		oveq1 7430	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑘) = (𝑁 − 𝑘))
40	39	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑁 − 𝑘)))
41	40	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))
42	38, 41	oveq12d 7441	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
43	42	adantr 486	. . . . . . 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 15783	. . . . . 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 2782	. . . . 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 343	. . . 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 16107	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
48		fallfac0 16107	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 FallFac 0) = 1)
49	47, 48	oveqan12d 7442	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = (1 · 1))
50		1t1e1 12420	. . . . . . . 8 ⊢ (1 · 1) = 1
51	49, 50	eqtrdi 2817	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = 1)
52	51	oveq2d 7439	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = (1 · 1))
53	52, 50	eqtrdi 2817	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = 1)
54		0cn 11216	. . . . . 6 ⊢ 0 ∈ ℂ
55		ax-1cn 11176	. . . . . . 7 ⊢ 1 ∈ ℂ
56	53, 55	eqeltrdi 2874	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ)
57		oveq2 7431	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
58		0nn0 12537	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
59		bcnn 14368	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
60	58, 59	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
61	57, 60	eqtrdi 2817	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
62		oveq2 7431	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
63		0m0e0 12377	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
64	62, 63	eqtrdi 2817	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
65	64	oveq2d 7439	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴 FallFac (0 − 𝑘)) = (𝐴 FallFac 0))
66		oveq2 7431	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵 FallFac 𝑘) = (𝐵 FallFac 0))
67	65, 66	oveq12d 7441	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac 0) · (𝐵 FallFac 0)))
68	61, 67	oveq12d 7441	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
69	68	sumsn 15823	. . . . . 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 599	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
71		addcl 11200	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
72		fallfac0 16107	. . . . . 6 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((𝐴 + 𝐵) FallFac 0) = 1)
73	71, 72	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = 1)
74	53, 70, 73	3eqtr4rd 2812	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
75		simprl 783	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
76		simprr 785	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
77		simpl 488	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ₀)
78		id 23	. . . . . . 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 16119	. . . . . 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 425	. . . . 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 30	. . . 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 12709	. . 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 33	. 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 1135	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 {csn 4594 (class class class)co 7423 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 − cmin 11459 ℕ₀cn0 12522 ...cfz 13553 Ccbc 14358 Σcsu 15763 FallFac cfallfac 16084

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 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 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-prod 15984 df-risefac 16086 df-fallfac 16087

This theorem is used by: binomrisefac 16121

W3C validator