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Theorem binomfallfac 14772

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 6658	. . . . . 6 ⊢ (𝑚 = 0 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 0))
2		oveq2 6658	. . . . . . . 8 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3		0z 11388	. . . . . . . . 9 ⊢ 0 ∈ ℤ
4		fzsn 12383	. . . . . . . . 9 ⊢ (0 ∈ ℤ → (0...0) = {0})
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (0...0) = {0}
6	2, 5	syl6eq 2672	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = {0})
7		oveq1 6657	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
8		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (𝑚 − 𝑘) = (0 − 𝑘))
9	8	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑚 = 0 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (0 − 𝑘)))
10	9	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑚 = 0 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)))
11	7, 10	oveq12d 6668	. . . . . . . 8 ⊢ (𝑚 = 0 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
12	11	adantr 481	. . . . . . 7 ⊢ ((𝑚 = 0 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
13	6, 12	sumeq12dv 14437	. . . . . 6 ⊢ (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
14	1, 13	eqeq12d 2637	. . . . 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)))))
15	14	imbi2d 330	. . . 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))))))
16		oveq2 6658	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑛))
17		oveq2 6658	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
18		oveq1 6657	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
19		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 − 𝑘) = (𝑛 − 𝑘))
20	19	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑛 − 𝑘)))
21	20	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘)))
22	18, 21	oveq12d 6668	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
23	22	adantr 481	. . . . . . 7 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑛C𝑘) · ((𝐴 FallFac (𝑛 − 𝑘)) · (𝐵 FallFac 𝑘))))
24	17, 23	sumeq12dv 14437	. . . . . 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))))
25	16, 24	eqeq12d 2637	. . . . 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)))))
26	25	imbi2d 330	. . . 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))))))
27		oveq2 6658	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac (𝑛 + 1)))
28		oveq2 6658	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
29		oveq1 6657	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
30		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 𝑘) = ((𝑛 + 1) − 𝑘))
31	30	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac ((𝑛 + 1) − 𝑘)))
32	31	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘)))
33	29, 32	oveq12d 6668	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴 FallFac ((𝑛 + 1) − 𝑘)) · (𝐵 FallFac 𝑘))))
34	33	adantr 481	. . . . . . 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))))
35	28, 34	sumeq12dv 14437	. . . . . 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))))
36	27, 35	eqeq12d 2637	. . . . 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)))))
37	36	imbi2d 330	. . . 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))))))
38		oveq2 6658	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 + 𝐵) FallFac 𝑚) = ((𝐴 + 𝐵) FallFac 𝑁))
39		oveq2 6658	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
40		oveq1 6657	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
41		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑘) = (𝑁 − 𝑘))
42	41	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (𝐴 FallFac (𝑚 − 𝑘)) = (𝐴 FallFac (𝑁 − 𝑘)))
43	42	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))
44	40, 43	oveq12d 6668	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
45	44	adantr 481	. . . . . . 7 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (0...𝑚)) → ((𝑚C𝑘) · ((𝐴 FallFac (𝑚 − 𝑘)) · (𝐵 FallFac 𝑘))) = ((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘))))
46	39, 45	sumeq12dv 14437	. . . . . 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))))
47	38, 46	eqeq12d 2637	. . . . 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)))))
48	47	imbi2d 330	. . . 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))))))
49		fallfac0 14759	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 FallFac 0) = 1)
50		fallfac0 14759	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 FallFac 0) = 1)
51	49, 50	oveqan12d 6669	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = (1 · 1))
52		1t1e1 11175	. . . . . . . 8 ⊢ (1 · 1) = 1
53	51, 52	syl6eq 2672	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 FallFac 0) · (𝐵 FallFac 0)) = 1)
54	53	oveq2d 6666	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = (1 · 1))
55	54, 52	syl6eq 2672	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) = 1)
56		0cn 10032	. . . . . 6 ⊢ 0 ∈ ℂ
57		ax-1cn 9994	. . . . . . 7 ⊢ 1 ∈ ℂ
58	55, 57	syl6eqel 2709	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))) ∈ ℂ)
59		oveq2 6658	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
60		0nn0 11307	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
61		bcnn 13099	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
62	60, 61	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
63	59, 62	syl6eq 2672	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
64		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
65		0m0e0 11130	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
66	64, 65	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
67	66	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴 FallFac (0 − 𝑘)) = (𝐴 FallFac 0))
68		oveq2 6658	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵 FallFac 𝑘) = (𝐵 FallFac 0))
69	67, 68	oveq12d 6668	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘)) = ((𝐴 FallFac 0) · (𝐵 FallFac 0)))
70	63, 69	oveq12d 6668	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
71	70	sumsn 14475	. . . . . 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))))
72	56, 58, 71	sylancr 695	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))) = (1 · ((𝐴 FallFac 0) · (𝐵 FallFac 0))))
73		addcl 10018	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
74		fallfac0 14759	. . . . . 6 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((𝐴 + 𝐵) FallFac 0) = 1)
75	73, 74	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = 1)
76	55, 72, 75	3eqtr4rd 2667	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 0) = Σ𝑘 ∈ {0} ((0C𝑘) · ((𝐴 FallFac (0 − 𝑘)) · (𝐵 FallFac 𝑘))))
77		simprl 794	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
78		simprr 796	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
79		simpl 473	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ₀)
80		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))))
81	77, 78, 79, 80	binomfallfaclem2 14771	. . . . . 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))))
82	81	exp31 630	. . . . 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))))))
83	82	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))))))
84	15, 26, 37, 48, 76, 83	nn0ind 11472	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))))
85	84	com12 32	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑁 ∈ ℕ₀ → ((𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 FallFac (𝑁 − 𝑘)) · (𝐵 FallFac 𝑘)))))
86	85	3impia 1261	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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {csn 4177 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 Ccbc 13089 Σcsu 14416 FallFac cfallfac 14735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-prod 14636 df-risefac 14737 df-fallfac 14738

This theorem is referenced by: binomrisefac 14773

W3C validator