Theorem bcxmas 15795

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 15794	. . 3 ⊢ (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2		oveq2 7368	. . . 4 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3	2	sumeq1d 15657	. . 3 ⊢ (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
4	1, 3	eqeq12d 2757	. 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 15794	. . 3 ⊢ (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
6		oveq2 7368	. . . 4 ⊢ (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
7	6	sumeq1d 15657	. . 3 ⊢ (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
8	5, 7	eqeq12d 2757	. 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 15794	. . 3 ⊢ (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
10		oveq2 7368	. . . 4 ⊢ (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
11	10	sumeq1d 15657	. . 3 ⊢ (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
12	9, 11	eqeq12d 2757	. 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 15794	. . 3 ⊢ (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
14		oveq2 7368	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
15	14	sumeq1d 15657	. . 3 ⊢ (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
16	13, 15	eqeq12d 2757	. 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 12447	. . . 4 ⊢ 0 ∈ ℕ0
18		nn0addcl 12467	. . . . 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 698	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
22		0z 12530	. . . 4 ⊢ 0 ∈ ℤ
23		1nn0 12448	. . . . . 6 ⊢ 1 ∈ ℕ0
24	21, 23	eqeltrdi 2849	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
25	24	nn0cnd 12495	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
26		bcxmaslem1 15794	. . . . 5 ⊢ (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
27	26	fsum1 15704	. . . 4 ⊢ ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
28	22, 25, 27	sylancr 594	. . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
29		peano2nn0 12472	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
30		nn0addcl 12467	. . . . 5 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
31	29, 17, 30	sylancl 593	. . . 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 2787	. 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
35		elnn0uz 12824	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
36	35	bilani 506	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
37		simpl 484	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
38		elfznn0 13569	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
39		nn0addcl 12467	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
40	37, 38, 39	syl2an 603	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
41		elfzelz 13473	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
42	41	adantl 483	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
43		bccl 14279	. . . . . . . 8 ⊢ (((𝑁 + 𝑗) ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
44	40, 42, 43	syl2anc 591	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
45	44	nn0cnd 12495	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
46		bcxmaslem1 15794	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
47	36, 45, 46	fsump1 15713	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))))
48		nn0cn 12442	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
49	48	adantr 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
50		nn0cn 12442	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
51	50	adantl 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
52		1cnd 11134	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℂ)
53		add32r 11361	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
54	49, 51, 52, 53	syl3anc 1380	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
55	54	oveq1d 7375	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
56	55	oveq2d 7376	. . . . 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))))
57	47, 56	eqtrd 2776	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
58	57	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))))
59		oveq1 7367	. . . 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))))
60	59	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))))
61		ax-1cn 11091	. . . . . . . . 9 ⊢ 1 ∈ ℂ
62		pncan 11394	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
63	51, 61, 62	sylancl 593	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
64	63	oveq2d 7376	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
65	64	oveq2d 7376	. . . . . 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)))
66		nn0addcl 12467	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
67	29, 66	sylan 587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
68		nn0p1nn 12471	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
69	68	adantl 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
70	69	nnzd 12545	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
71		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)))
72	67, 70, 71	syl2anc 591	. . . . . 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)))
73	65, 72	eqtr3d 2778	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
74		nn0p1nn 12471	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
75		nnnn0addcl 12462	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
76	74, 75	sylan 587	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
77	76	nnnn0d 12493	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
78		bccl 14279	. . . . . . . 8 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
79	77, 70, 78	syl2anc 591	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
80	79	nn0cnd 12495	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
81		nn0z 12543	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
82	81	adantl 483	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
83		bccl 14279	. . . . . . . . 9 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
84	66, 82, 83	syl2anc 591	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
85	29, 84	sylan 587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
86	85	nn0cnd 12495	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
87	80, 86	addcomd 11343	. . . . 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))))
88		peano2cn 11313	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
89	48, 88	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
90	89	adantr 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
91	90, 51, 52	addassd 11162	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
92	91	oveq1d 7375	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
93	73, 87, 92	3eqtr3d 2784	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
94	93	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)))
95	58, 60, 94	3eqtr2rd 2783	. 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))
96	4, 8, 12, 16, 34, 95	nn0indd 12621	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ‘cfv 6489  (class class class)co 7360  ℂcc 11031  0cc0 11033  1c1 11034   + caddc 11036   − cmin 11372  ℕcn 12169  ℕ₀cn0 12432  ℤcz 12519  ℤ_≥cuz 12783  ...cfz 13456  Ccbc 14259  Σcsu 15643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-oi 9419  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019  df-fac 14231  df-bc 14260  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-sum 15644

This theorem is referenced by:  arisum  15820