Theorem bccl 10513

Description: A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)

Assertion

Ref	Expression
bccl	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)

Proof of Theorem bccl

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

Step	Hyp	Ref	Expression
1		oveq1 5781	. . . . 5 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
2	1	eleq1d 2208	. . . 4 ⊢ (𝑚 = 0 → ((𝑚C𝑘) ∈ ℕ0 ↔ (0C𝑘) ∈ ℕ0))
3	2	ralbidv 2437	. . 3 ⊢ (𝑚 = 0 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0))
4		oveq1 5781	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
5	4	eleq1d 2208	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
6	5	ralbidv 2437	. . 3 ⊢ (𝑚 = 𝑛 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0))
7		oveq1 5781	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
8	7	eleq1d 2208	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚C𝑘) ∈ ℕ0 ↔ ((𝑛 + 1)C𝑘) ∈ ℕ0))
9	8	ralbidv 2437	. . 3 ⊢ (𝑚 = (𝑛 + 1) → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
10		oveq1 5781	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
11	10	eleq1d 2208	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑁C𝑘) ∈ ℕ0))
12	11	ralbidv 2437	. . 3 ⊢ (𝑚 = 𝑁 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0))
13		elfz1eq 9815	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
14	13	adantl 275	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
15		oveq2 5782	. . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
16		0nn0 8992	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
17		bcn0 10501	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
19		1nn0 8993	. . . . . . . 8 ⊢ 1 ∈ ℕ0
20	18, 19	eqeltri 2212	. . . . . . 7 ⊢ (0C0) ∈ ℕ0
21	15, 20	eqeltrdi 2230	. . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) ∈ ℕ0)
22	14, 21	syl 14	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
23		bcval3 10497	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
24	16, 23	mp3an1 1302	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
25	24, 16	eqeltrdi 2230	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
26		0zd 9066	. . . . . 6 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
27		fzdcel 9820	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑘 ∈ (0...0))
28		exmiddc 821	. . . . . . 7 ⊢ (DECID 𝑘 ∈ (0...0) → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
29	27, 28	syl 14	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
30	26, 26, 29	mpd3an23 1317	. . . . 5 ⊢ (𝑘 ∈ ℤ → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
31	22, 25, 30	mpjaodan 787	. . . 4 ⊢ (𝑘 ∈ ℤ → (0C𝑘) ∈ ℕ0)
32	31	rgen 2485	. . 3 ⊢ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0
33		oveq2 5782	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑛C𝑘) = (𝑛C𝑚))
34	33	eleq1d 2208	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝑛C𝑘) ∈ ℕ0 ↔ (𝑛C𝑚) ∈ ℕ0))
35	34	cbvralv 2654	. . . 4 ⊢ (∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0 ↔ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0)
36		bcpasc 10512	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
37	36	adantlr 468	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
38		oveq2 5782	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑛C𝑚) = (𝑛C𝑘))
39	38	eleq1d 2208	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
40	39	rspccva 2788	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑛C𝑘) ∈ ℕ0)
41		peano2zm 9092	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
42		oveq2 5782	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (𝑛C𝑚) = (𝑛C(𝑘 − 1)))
43	42	eleq1d 2208	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C(𝑘 − 1)) ∈ ℕ0))
44	43	rspccva 2788	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ (𝑘 − 1) ∈ ℤ) → (𝑛C(𝑘 − 1)) ∈ ℕ0)
45	41, 44	sylan2 284	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑛C(𝑘 − 1)) ∈ ℕ0)
46	40, 45	nn0addcld 9034	. . . . . . . 8 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
47	46	adantll 467	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
48	37, 47	eqeltrrd 2217	. . . . . 6 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛 + 1)C𝑘) ∈ ℕ0)
49	48	ralrimiva 2505	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0)
50	49	ex 114	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
51	35, 50	syl5bi 151	. . 3 ⊢ (𝑛 ∈ ℕ0 → (∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0 → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
52	3, 6, 9, 12, 32, 51	nn0ind 9165	. 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0)
53		oveq2 5782	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑁C𝑘) = (𝑁C𝐾))
54	53	eleq1d 2208	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑁C𝑘) ∈ ℕ0 ↔ (𝑁C𝐾) ∈ ℕ0))
55	54	rspccva 2788	. 2 ⊢ ((∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)
56	52, 55	sylan 281	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416  (class class class)co 5774  0cc0 7620  1c1 7621   + caddc 7623   − cmin 7933  ℕ₀cn0 8977  ℤcz 9054  ...cfz 9790  Ccbc 10493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-seqfrec 10219  df-fac 10472  df-bc 10494

This theorem is referenced by:  bccl2  10514  bcn2m1  10515  bcn2p1  10516  binomlem  11252  bcxmas  11258