Theorem bcxmas 12175

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 12174	. . . . 5 ⊢ (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2		oveq2 6058	. . . . . 6 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
3	2	sumeq1d 12051	. . . . 5 ⊢ (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
4	1, 3	eqeq12d 2247	. . . 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 230	. . 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 12174	. . . . 5 ⊢ (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
7		oveq2 6058	. . . . . 6 ⊢ (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
8	7	sumeq1d 12051	. . . . 5 ⊢ (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
9	6, 8	eqeq12d 2247	. . . 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 230	. . 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 12174	. . . . 5 ⊢ (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
12		oveq2 6058	. . . . . 6 ⊢ (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
13	12	sumeq1d 12051	. . . . 5 ⊢ (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
14	11, 13	eqeq12d 2247	. . . 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 230	. . 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 12174	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
17		oveq2 6058	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
18	17	sumeq1d 12051	. . . . 5 ⊢ (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
19	16, 18	eqeq12d 2247	. . . 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 230	. . 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 9511	. . . . 5 ⊢ 0 ∈ ℕ0
22		nn0addcl 9531	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
23		bcn0 11117	. . . . . 6 ⊢ ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
24	22, 23	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
25	21, 24	mpan2 425	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
26		0z 9588	. . . . 5 ⊢ 0 ∈ ℤ
27		1nn0 9512	. . . . . . 7 ⊢ 1 ∈ ℕ0
28	25, 27	eqeltrdi 2323	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
29	28	nn0cnd 9555	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
30		bcxmaslem1 12174	. . . . . 6 ⊢ (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
31	30	fsum1 12098	. . . . 5 ⊢ ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
32	26, 29, 31	sylancr 414	. . . 4 ⊢ (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
33		peano2nn0 9536	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
34		nn0addcl 9531	. . . . . 6 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
35	33, 21, 34	sylancl 413	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
36		bcn0 11117	. . . . 5 ⊢ (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
37	35, 36	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
38	25, 32, 37	3eqtr4rd 2276	. . 3 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
39		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
40		elnn0uz 9892	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
41	39, 40	sylib 122	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
42		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
43		elfznn0 10448	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
44		nn0addcl 9531	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
45	42, 43, 44	syl2an 289	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
46		elfzelz 10359	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
47	46	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
48		bccl 11129	. . . . . . . . . . . 12 ⊢ (((𝑁 + 𝑗) ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
49	45, 47, 48	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
50	49	nn0cnd 9555	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
51		bcxmaslem1 12174	. . . . . . . . . 10 ⊢ (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
52	41, 50, 51	fsump1 12106	. . . . . . . . 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 9506	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
54	53	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
55		nn0cn 9506	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
56	55	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
57		1cnd 8290	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℂ)
58		add32r 8433	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
59	54, 56, 57, 58	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
60	59	oveq1d 6065	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
61	60	oveq2d 6066	. . . . . . . . 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 2265	. . . . . . . 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 276	. . . . . . 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 6057	. . . . . . . 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 277	. . . . . . 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 8220	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
67		pncan 8479	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
68	56, 66, 67	sylancl 413	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
69	68	oveq2d 6066	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
70	69	oveq2d 6066	. . . . . . . . . 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 9531	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
72	33, 71	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
73		nn0p1nn 9535	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
74	73	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
75	74	nnzd 9699	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
76		bcpasc 11128	. . . . . . . . . . 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 411	. . . . . . . . . 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 2267	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
79		nn0p1nn 9535	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
80		nnnn0addcl 9526	. . . . . . . . . . . . . 14 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
81	79, 80	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
82	81	nnnn0d 9553	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
83		bccl 11129	. . . . . . . . . . . 12 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
84	82, 75, 83	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
85	84	nn0cnd 9555	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
86		nn0z 9597	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
87	86	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
88		bccl 11129	. . . . . . . . . . . . 13 ⊢ ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
89	71, 87, 88	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
90	33, 89	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
91	90	nn0cnd 9555	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
92	85, 91	addcomd 8424	. . . . . . . . 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 8408	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
94	53, 93	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
95	94	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
96	95, 56, 57	addassd 8296	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
97	96	oveq1d 6065	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
98	78, 92, 97	3eqtr3d 2273	. . . . . . . 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 276	. . . . . . 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 2272	. . . . . 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 115	. . . . 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 116	. . . 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 26	. . 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 9692	. 2 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
105	104	impcom 125	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ‘cfv 5352  (class class class)co 6050  ℂcc 8125  0cc0 8127  1c1 8128   + caddc 8130   − cmin 8444  ℕcn 9237  ℕ₀cn0 9496  ℤcz 9577  ℤ_≥cuz 9853  ...cfz 10342  Ccbc 11109  Σcsu 12038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-bc 11110  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  arisum  12184