Theorem bcxmas 14567

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 14566	. . . . 5 ⊢ (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2		oveq2 6658	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3	2	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
4	1, 3	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
5	4	imbi2d 330	. . 3 ⊢ (𝑚 = 0 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))))
6		bcxmaslem1 14566	. . . . 5 ⊢ (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7		oveq2 6658	. . . . . 6 ⊢ (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
8	7	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
9	6, 8	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
10	9	imbi2d 330	. . 3 ⊢ (𝑚 = 𝑘 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))))
11		bcxmaslem1 14566	. . . . 5 ⊢ (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12		oveq2 6658	. . . . . 6 ⊢ (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
13	12	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
14	11, 13	eqeq12d 2637	. . . 4 ⊢ (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)))
15	14	imbi2d 330	. . 3 ⊢ (𝑚 = (𝑘 + 1) → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))))
16		bcxmaslem1 14566	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17		oveq2 6658	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
18	17	sumeq1d 14431	. . . . 5 ⊢ (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
19	16, 18	eqeq12d 2637	. . . 4 ⊢ (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
20	19	imbi2d 330	. . 3 ⊢ (𝑚 = 𝑀 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗)) ↔ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))))
21		0nn0 11307	. . . . 5 ⊢ 0 ∈ ℕ0
22		nn0addcl 11328	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23		bcn0 13097	. . . . . 6 ⊢ ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
24	22, 23	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
25	21, 24	mpan2 707	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26		0z 11388	. . . . 5 ⊢ 0 ∈ ℤ
27		1nn0 11308	. . . . . . 7 ⊢ 1 ∈ ℕ0
28	25, 27	syl6eqel 2709	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
29	28	nn0cnd 11353	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30		bcxmaslem1 14566	. . . . . 6 ⊢ (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
31	30	fsum1 14476	. . . . 5 ⊢ ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
32	26, 29, 31	sylancr 695	. . . 4 ⊢ (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33		peano2nn0 11333	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34		nn0addcl 11328	. . . . . 6 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
35	33, 21, 34	sylancl 694	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36		bcn0 13097	. . . . 5 ⊢ (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
37	35, 36	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
38	25, 32, 37	3eqtr4rd 2667	. . 3 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40		elnn0uz 11725	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
41	39, 40	sylib 208	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
42		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43		elfznn0 12433	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44		nn0addcl 11328	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
45	42, 43, 44	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46		elfzelz 12342	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
47	46	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48		bccl 13109	. . . . . . . . . . . 12 ⊢ (((𝑁 + 𝑗) ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
49	45, 47, 48	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
50	49	nn0cnd 11353	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51		bcxmaslem1 14566	. . . . . . . . . 10 ⊢ (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
52	41, 50, 51	fsump1 14487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))))
53		nn0cn 11302	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
54	53	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55		nn0cn 11302	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
56	55	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57		1cnd 10056	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58		add32r 10255	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
59	54, 56, 57, 58	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
60	59	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
61	60	oveq2d 6666	. . . . . . . . 9 ⊢ ((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))))
62	52, 61	eqtrd 2656	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
63	62	adantr 481	. . . . . . 7 ⊢ (((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))))
64		oveq1 6657	. . . . . . . 8 ⊢ ((((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))))
65	64	adantl 482	. . . . . . 7 ⊢ (((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))))
66		ax-1cn 9994	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
67		pncan 10287	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
68	56, 66, 67	sylancl 694	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
69	68	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
70	69	oveq2d 6666	. . . . . . . . . 10 ⊢ ((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)))
71		nn0addcl 11328	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
72	33, 71	sylan 488	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73		nn0p1nn 11332	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
74	73	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
75	74	nnzd 11481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76		bcpasc 13108	. . . . . . . . . . 11 ⊢ ((((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)))
77	72, 75, 76	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
78	70, 77	eqtr3d 2658	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79		nn0p1nn 11332	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80		nnnn0addcl 11323	. . . . . . . . . . . . . 14 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
81	79, 80	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
82	81	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83		bccl 13109	. . . . . . . . . . . 12 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
84	82, 75, 83	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
85	84	nn0cnd 11353	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86		nn0z 11400	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
87	86	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88		bccl 13109	. . . . . . . . . . . . 13 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
89	71, 87, 88	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
90	33, 89	sylan 488	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
91	90	nn0cnd 11353	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
92	85, 91	addcomd 10238	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
93		peano2cn 10208	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
94	53, 93	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
95	94	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
96	95, 56, 57	addassd 10062	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
97	96	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
98	78, 92, 97	3eqtr3d 2664	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
99	98	adantr 481	. . . . . . 7 ⊢ (((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)))
100	63, 65, 99	3eqtr2rd 2663	. . . . . 6 ⊢ (((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))
101	100	ex 450	. . . . 5 ⊢ ((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)))
102	101	expcom 451	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝑁 ∈ 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))))
103	102	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))))
104	5, 10, 15, 20, 38, 103	nn0ind 11472	. 2 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105	104	impcom 446	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  Ccbc 13089  Σcsu 14416

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

This theorem is referenced by:  arisum  14592