Theorem bccl 10738

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 5877	. . . . 5 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
2	1	eleq1d 2246	. . . 4 ⊢ (𝑚 = 0 → ((𝑚C𝑘) ∈ ℕ0 ↔ (0C𝑘) ∈ ℕ0))
3	2	ralbidv 2477	. . 3 ⊢ (𝑚 = 0 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0))
4		oveq1 5877	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
5	4	eleq1d 2246	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
6	5	ralbidv 2477	. . 3 ⊢ (𝑚 = 𝑛 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0))
7		oveq1 5877	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
8	7	eleq1d 2246	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚C𝑘) ∈ ℕ0 ↔ ((𝑛 + 1)C𝑘) ∈ ℕ0))
9	8	ralbidv 2477	. . 3 ⊢ (𝑚 = (𝑛 + 1) → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
10		oveq1 5877	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
11	10	eleq1d 2246	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑁C𝑘) ∈ ℕ0))
12	11	ralbidv 2477	. . 3 ⊢ (𝑚 = 𝑁 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0))
13		elfz1eq 10028	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
14	13	adantl 277	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
15		oveq2 5878	. . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
16		0nn0 9185	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
17		bcn0 10726	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
19		1nn0 9186	. . . . . . . 8 ⊢ 1 ∈ ℕ0
20	18, 19	eqeltri 2250	. . . . . . 7 ⊢ (0C0) ∈ ℕ0
21	15, 20	eqeltrdi 2268	. . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) ∈ ℕ0)
22	14, 21	syl 14	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
23		bcval3 10722	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
24	16, 23	mp3an1 1324	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
25	24, 16	eqeltrdi 2268	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
26		0zd 9259	. . . . . 6 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
27		fzdcel 10033	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑘 ∈ (0...0))
28		exmiddc 836	. . . . . . 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 1339	. . . . 5 ⊢ (𝑘 ∈ ℤ → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
31	22, 25, 30	mpjaodan 798	. . . 4 ⊢ (𝑘 ∈ ℤ → (0C𝑘) ∈ ℕ0)
32	31	rgen 2530	. . 3 ⊢ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0
33		oveq2 5878	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑛C𝑘) = (𝑛C𝑚))
34	33	eleq1d 2246	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝑛C𝑘) ∈ ℕ0 ↔ (𝑛C𝑚) ∈ ℕ0))
35	34	cbvralv 2703	. . . 4 ⊢ (∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0 ↔ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0)
36		bcpasc 10737	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
37	36	adantlr 477	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
38		oveq2 5878	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑛C𝑚) = (𝑛C𝑘))
39	38	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
40	39	rspccva 2840	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑛C𝑘) ∈ ℕ0)
41		peano2zm 9285	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
42		oveq2 5878	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (𝑛C𝑚) = (𝑛C(𝑘 − 1)))
43	42	eleq1d 2246	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C(𝑘 − 1)) ∈ ℕ0))
44	43	rspccva 2840	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ (𝑘 − 1) ∈ ℤ) → (𝑛C(𝑘 − 1)) ∈ ℕ0)
45	41, 44	sylan2 286	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑛C(𝑘 − 1)) ∈ ℕ0)
46	40, 45	nn0addcld 9227	. . . . . . . 8 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
47	46	adantll 476	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
48	37, 47	eqeltrrd 2255	. . . . . 6 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛 + 1)C𝑘) ∈ ℕ0)
49	48	ralrimiva 2550	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0)
50	49	ex 115	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
51	35, 50	biimtrid 152	. . 3 ⊢ (𝑛 ∈ ℕ0 → (∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0 → ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
52	3, 6, 9, 12, 32, 51	nn0ind 9361	. 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0)
53		oveq2 5878	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑁C𝑘) = (𝑁C𝐾))
54	53	eleq1d 2246	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑁C𝑘) ∈ ℕ0 ↔ (𝑁C𝐾) ∈ ℕ0))
55	54	rspccva 2840	. 2 ⊢ ((∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)
56	52, 55	sylan 283	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) ∈ ℕ0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  (class class class)co 5870  0cc0 7806  1c1 7807   + caddc 7809   − cmin 8122  ℕ₀cn0 9170  ℤcz 9247  ...cfz 10002  Ccbc 10718

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-precex 7916  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922  ax-pre-mulgt0 7923  ax-pre-mulext 7924

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-frec 6387  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-reap 8526  df-ap 8533  df-div 8624  df-inn 8914  df-n0 9171  df-z 9248  df-uz 9523  df-q 9614  df-rp 9648  df-fz 10003  df-seqfrec 10439  df-fac 10697  df-bc 10719

This theorem is referenced by:  bccl2  10739  bcn2m1  10740  bcn2p1  10741  binomlem  11482  bcxmas  11488