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Theorem binomrisefac 16220

Description: A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)

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

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑁

**Proof of Theorem binomrisefac**
Step	Hyp	Ref	Expression
1		negdi 11625	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = ( -𝐴 + -𝐵))
2	1	3adant3 1150	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -(𝐴 + 𝐵) = ( -𝐴 + -𝐵))
3	2	oveq1d 7442	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ( -(𝐴 + 𝐵) FallFac 𝑁) = (( -𝐴 + -𝐵) FallFac 𝑁))
4		negcl 11567	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
5		negcl 11567	. . . . . 6 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ)
6		id 23	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
7		binomfallfac 16219	. . . . . 6 ⊢ (( -𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (( -𝐴 + -𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
8	4, 5, 6, 7	syl3an 1178	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (( -𝐴 + -𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
9	3, 8	eqtrd 2798	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ( -(𝐴 + 𝐵) FallFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
10	9	oveq2d 7443	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (( -1↑𝑁) · ( -(𝐴 + 𝐵) FallFac 𝑁)) = (( -1↑𝑁) · Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
11		fzfid 14126	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (0...𝑁) ∈ Fin)
12		neg1cn 12315	. . . . . 6 ⊢ -1 ∈ ℂ
13		expcl 14232	. . . . . 6 ⊢ (( -1 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ( -1↑𝑁) ∈ ℂ)
14	12, 13	mpan 703	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ( -1↑𝑁) ∈ ℂ)
15	14	3ad2ant3 1153	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ( -1↑𝑁) ∈ ℂ)
16		simp3 1156	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
17		elfzelz 13666	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ)
18		bccl 14476	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ₀)
19	16, 17, 18	syl2an 608	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ₀)
20	19	nn0cnd 12679	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ)
21		simpl1 1210	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)
22	21	negcld 11666	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -𝐴 ∈ ℂ)
23	16	nn0zd 12728	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
24		zsubcl 12748	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ)
25	23, 17, 24	syl2an 608	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ)
26		elfzle2 13671	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ≤ 𝑁)
27	26	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≤ 𝑁)
28		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
29	28	nn0red 12678	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
30		elfznn0 13764	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
31	30	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
32	31	nn0red 12678	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℝ)
33	29, 32	subge0d 11916	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (0 ≤ (𝑁 − 𝑘) ↔ 𝑘 ≤ 𝑁))
34	27, 33	mpbird 260	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 0 ≤ (𝑁 − 𝑘))
35		elnn0z 12716	. . . . . . . 8 ⊢ ((𝑁 − 𝑘) ∈ ℕ₀ ↔ ((𝑁 − 𝑘) ∈ ℤ ∧ 0 ≤ (𝑁 − 𝑘)))
36	25, 34, 35	sylanbrc 595	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℕ₀)
37		fallfaccl 16195	. . . . . . 7 ⊢ (( -𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → ( -𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
38	22, 36, 37	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -𝐴 FallFac (𝑁 − 𝑘)) ∈ ℂ)
39		simp2 1155	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝐵 ∈ ℂ)
40	39	negcld 11666	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → -𝐵 ∈ ℂ)
41		fallfaccl 16195	. . . . . . 7 ⊢ (( -𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ( -𝐵 FallFac 𝑘) ∈ ℂ)
42	40, 30, 41	syl2an 608	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -𝐵 FallFac 𝑘) ∈ ℂ)
43	38, 42	mulcld 11339	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)) ∈ ℂ)
44	20, 43	mulcld 11339	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))) ∈ ℂ)
45	11, 15, 44	fsummulc2 15962	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (( -1↑𝑁) · Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))) = Σ𝑘 ∈ (0...𝑁)(( -1↑𝑁) · ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
46	10, 45	eqtrd 2798	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (( -1↑𝑁) · ( -(𝐴 + 𝐵) FallFac 𝑁)) = Σ𝑘 ∈ (0...𝑁)(( -1↑𝑁) · ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
47		addcl 11292	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
48		risefallfac 16203	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = (( -1↑𝑁) · ( -(𝐴 + 𝐵) FallFac 𝑁)))
49	47, 48	stoic3 1808	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = (( -1↑𝑁) · ( -(𝐴 + 𝐵) FallFac 𝑁)))
50		risefallfac 16203	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → (𝐴 RiseFac (𝑁 − 𝑘)) = (( -1↑(𝑁 − 𝑘)) · ( -𝐴 FallFac (𝑁 − 𝑘))))
51	21, 36, 50	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴 RiseFac (𝑁 − 𝑘)) = (( -1↑(𝑁 − 𝑘)) · ( -𝐴 FallFac (𝑁 − 𝑘))))
52		simpl2 1211	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ)
53		risefallfac 16203	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵 RiseFac 𝑘) = (( -1↑𝑘) · ( -𝐵 FallFac 𝑘)))
54	52, 31, 53	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (𝐵 RiseFac 𝑘) = (( -1↑𝑘) · ( -𝐵 FallFac 𝑘)))
55	51, 54	oveq12d 7445	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘)) = ((( -1↑(𝑁 − 𝑘)) · ( -𝐴 FallFac (𝑁 − 𝑘))) · (( -1↑𝑘) · ( -𝐵 FallFac 𝑘))))
56		expcl 14232	. . . . . . . 8 ⊢ (( -1 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ₀) → ( -1↑(𝑁 − 𝑘)) ∈ ℂ)
57	12, 36, 56	sylancr 599	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -1↑(𝑁 − 𝑘)) ∈ ℂ)
58		expcl 14232	. . . . . . . . 9 ⊢ (( -1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ( -1↑𝑘) ∈ ℂ)
59	12, 30, 58	sylancr 599	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → ( -1↑𝑘) ∈ ℂ)
60	59	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -1↑𝑘) ∈ ℂ)
61	57, 38, 60, 42	mul4d 11532	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((( -1↑(𝑁 − 𝑘)) · ( -𝐴 FallFac (𝑁 − 𝑘))) · (( -1↑𝑘) · ( -𝐵 FallFac 𝑘))) = ((( -1↑(𝑁 − 𝑘)) · ( -1↑𝑘)) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
62	12	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → -1 ∈ ℂ)
63	62, 31, 36	expaddd 14301	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -1↑((𝑁 − 𝑘) + 𝑘)) = (( -1↑(𝑁 − 𝑘)) · ( -1↑𝑘)))
64	16	nn0cnd 12679	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℂ)
65	30	nn0cnd 12679	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℂ)
66		npcan 11576	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
67	64, 65, 66	syl2an 608	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 − 𝑘) + 𝑘) = 𝑁)
68	67	oveq2d 7443	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -1↑((𝑁 − 𝑘) + 𝑘)) = ( -1↑𝑁))
69	63, 68	eqtr3d 2800	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → (( -1↑(𝑁 − 𝑘)) · ( -1↑𝑘)) = ( -1↑𝑁))
70	69	oveq1d 7442	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((( -1↑(𝑁 − 𝑘)) · ( -1↑𝑘)) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))) = (( -1↑𝑁) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
71	55, 61, 70	3eqtrd 2802	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘)) = (( -1↑𝑁) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘))))
72	71	oveq2d 7443	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = ((𝑁C𝑘) · (( -1↑𝑁) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
73	15	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ( -1↑𝑁) ∈ ℂ)
74	20, 73, 43	mul12d 11529	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (( -1↑𝑁) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))) = (( -1↑𝑁) · ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
75	72, 74	eqtrd 2798	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = (( -1↑𝑁) · ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
76	75	sumeq2dv 15881	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))) = Σ𝑘 ∈ (0...𝑁)(( -1↑𝑁) · ((𝑁C𝑘) · (( -𝐴 FallFac (𝑁 − 𝑘)) · ( -𝐵 FallFac 𝑘)))))
77	46, 49, 76	3eqtr4d 2808	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵) RiseFac 𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴 RiseFac (𝑁 − 𝑘)) · (𝐵 RiseFac 𝑘))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 class class class wbr 5105 (class class class)co 7427 ℂcc 11208 0cc0 11210 1c1 11211 + caddc 11213 · cmul 11215 ≤ cle 11354 − cmin 11551 -cneg 11552 ℕ₀cn0 12616 ℤcz 12703 ...cfz 13649 ↑cexp 14214 Ccbc 14456 Σcsu 15865 FallFac cfallfac 16183 RiseFac crisefac 16184

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-inf2 9652 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9444 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-n0 12617 df-z 12704 df-uz 12976 df-rp 13131 df-fz 13650 df-fzo 13799 df-seq 14155 df-exp 14215 df-fac 14428 df-bc 14457 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-clim 15667 df-sum 15866 df-prod 16085 df-risefac 16185 df-fallfac 16186

This theorem is used by: (None)

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