Theorem bcxmas 15778

Description: Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)

Assertion

Ref	Expression
bcxmas	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

Distinct variable groups:   𝑗,𝑀   𝑗,𝑁

Proof of Theorem bcxmas

Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcxmaslem1 15777	. . 3 ⊢ (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2		oveq2 7414	. . . 4 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3	2	sumeq1d 15644	. . 3 ⊢ (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
4	1, 3	eqeq12d 2749	. 2 ⊢ (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
5		bcxmaslem1 15777	. . 3 ⊢ (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
6		oveq2 7414	. . . 4 ⊢ (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
7	6	sumeq1d 15644	. . 3 ⊢ (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
8	5, 7	eqeq12d 2749	. 2 ⊢ (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
9		bcxmaslem1 15777	. . 3 ⊢ (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
10		oveq2 7414	. . . 4 ⊢ (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
11	10	sumeq1d 15644	. . 3 ⊢ (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
12	9, 11	eqeq12d 2749	. 2 ⊢ (𝑚 = (𝑘 + 1) → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗)))
13		bcxmaslem1 15777	. . 3 ⊢ (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
14		oveq2 7414	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
15	14	sumeq1d 15644	. . 3 ⊢ (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
16	13, 15	eqeq12d 2749	. 2 ⊢ (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
17		0nn0 12484	. . . 4 ⊢ 0 ∈ ℕ0
18		nn0addcl 12504	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
19		bcn0 14267	. . . . 5 ⊢ ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
20	18, 19	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
21	17, 20	mpan2 690	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
22		0z 12566	. . . 4 ⊢ 0 ∈ ℤ
23		1nn0 12485	. . . . . 6 ⊢ 1 ∈ ℕ0
24	21, 23	eqeltrdi 2842	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
25	24	nn0cnd 12531	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
26		bcxmaslem1 15777	. . . . 5 ⊢ (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
27	26	fsum1 15690	. . . 4 ⊢ ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
28	22, 25, 27	sylancr 588	. . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
29		peano2nn0 12509	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
30		nn0addcl 12504	. . . . 5 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
31	29, 17, 30	sylancl 587	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
32		bcn0 14267	. . . 4 ⊢ (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
33	31, 32	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
34	21, 28, 33	3eqtr4rd 2784	. 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
35		simpr 486	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
36		elnn0uz 12864	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
37	35, 36	sylib 217	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
38		simpl 484	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
39		elfznn0 13591	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
40		nn0addcl 12504	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
41	38, 39, 40	syl2an 597	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
42		elfzelz 13498	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
43	42	adantl 483	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
44		bccl 14279	. . . . . . . 8 ⊢ (((𝑁 + 𝑗) ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
45	41, 43, 44	syl2anc 585	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
46	45	nn0cnd 12531	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
47		bcxmaslem1 15777	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
48	37, 46, 47	fsump1 15699	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))))
49		nn0cn 12479	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
50	49	adantr 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
51		nn0cn 12479	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
52	51	adantl 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
53		1cnd 11206	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℂ)
54		add32r 11430	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
55	50, 52, 53, 54	syl3anc 1372	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
56	55	oveq1d 7421	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
57	56	oveq2d 7422	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
58	48, 57	eqtrd 2773	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
59	58	adantr 482	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
60		oveq1 7413	. . . 4 ⊢ ((((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
61	60	adantl 483	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
62		ax-1cn 11165	. . . . . . . . 9 ⊢ 1 ∈ ℂ
63		pncan 11463	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
64	52, 62, 63	sylancl 587	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
65	64	oveq2d 7422	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
66	65	oveq2d 7422	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)))
67		nn0addcl 12504	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
68	29, 67	sylan 581	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
69		nn0p1nn 12508	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
70	69	adantl 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
71	70	nnzd 12582	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
72		bcpasc 14278	. . . . . . 7 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
73	68, 71, 72	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
74	66, 73	eqtr3d 2775	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
75		nn0p1nn 12508	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
76		nnnn0addcl 12499	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
77	75, 76	sylan 581	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
78	77	nnnn0d 12529	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
79		bccl 14279	. . . . . . . 8 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
80	78, 71, 79	syl2anc 585	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
81	80	nn0cnd 12531	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
82		nn0z 12580	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
83	82	adantl 483	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
84		bccl 14279	. . . . . . . . 9 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
85	67, 83, 84	syl2anc 585	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
86	29, 85	sylan 581	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
87	86	nn0cnd 12531	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
88	81, 87	addcomd 11413	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
89		peano2cn 11383	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
90	49, 89	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
91	90	adantr 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
92	91, 52, 53	addassd 11233	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
93	92	oveq1d 7421	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
94	74, 88, 93	3eqtr3d 2781	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
95	94	adantr 482	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
96	59, 61, 95	3eqtr2rd 2780	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
97	4, 8, 12, 16, 34, 96	nn0indd 12656	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ‘cfv 6541  (class class class)co 7406  ℂcc 11105  0cc0 11107  1c1 11108   + caddc 11110   − cmin 11441  ℕcn 12209  ℕ₀cn0 12469  ℤcz 12555  ℤ_≥cuz 12819  ...cfz 13481  Ccbc 14259  Σcsu 15629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-seq 13964  df-exp 14025  df-fac 14231  df-bc 14260  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630

This theorem is referenced by:  arisum  15803