Theorem bccl 11188

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 6086	. . . . 5 ⊢ (𝑚 = 0 → (𝑚C𝑘) = (0C𝑘))
2	1	eleq1d 2307	. . . 4 ⊢ (𝑚 = 0 → ((𝑚C𝑘) ∈ ℕ0 ↔ (0C𝑘) ∈ ℕ0))
3	2	ralbidv 2550	. . 3 ⊢ (𝑚 = 0 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0))
4		oveq1 6086	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚C𝑘) = (𝑛C𝑘))
5	4	eleq1d 2307	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
6	5	ralbidv 2550	. . 3 ⊢ (𝑚 = 𝑛 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0))
7		oveq1 6086	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚C𝑘) = ((𝑛 + 1)C𝑘))
8	7	eleq1d 2307	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚C𝑘) ∈ ℕ0 ↔ ((𝑛 + 1)C𝑘) ∈ ℕ0))
9	8	ralbidv 2550	. . 3 ⊢ (𝑚 = (𝑛 + 1) → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ ((𝑛 + 1)C𝑘) ∈ ℕ0))
10		oveq1 6086	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚C𝑘) = (𝑁C𝑘))
11	10	eleq1d 2307	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑚C𝑘) ∈ ℕ0 ↔ (𝑁C𝑘) ∈ ℕ0))
12	11	ralbidv 2550	. . 3 ⊢ (𝑚 = 𝑁 → (∀𝑘 ∈ ℤ (𝑚C𝑘) ∈ ℕ0 ↔ ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0))
13		elfz1eq 10422	. . . . . . 7 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
14	13	adantl 277	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
15		oveq2 6087	. . . . . . 7 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
16		0nn0 9561	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
17		bcn0 11176	. . . . . . . . 9 ⊢ (0 ∈ ℕ0 → (0C0) = 1)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (0C0) = 1
19		1nn0 9562	. . . . . . . 8 ⊢ 1 ∈ ℕ0
20	18, 19	eqeltri 2311	. . . . . . 7 ⊢ (0C0) ∈ ℕ0
21	15, 20	eqeltrdi 2329	. . . . . 6 ⊢ (𝑘 = 0 → (0C𝑘) ∈ ℕ0)
22	14, 21	syl 14	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
23		bcval3 11172	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
24	16, 23	mp3an1 1365	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
25	24, 16	eqeltrdi 2329	. . . . 5 ⊢ ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) ∈ ℕ0)
26		0zd 9639	. . . . . 6 ⊢ (𝑘 ∈ ℤ → 0 ∈ ℤ)
27		fzdcel 10427	. . . . . . 7 ⊢ ((𝑘 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑘 ∈ (0...0))
28		exmiddc 848	. . . . . . 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 1380	. . . . 5 ⊢ (𝑘 ∈ ℤ → (𝑘 ∈ (0...0) ∨ ¬ 𝑘 ∈ (0...0)))
31	22, 25, 30	mpjaodan 810	. . . 4 ⊢ (𝑘 ∈ ℤ → (0C𝑘) ∈ ℕ0)
32	31	rgen 2603	. . 3 ⊢ ∀𝑘 ∈ ℤ (0C𝑘) ∈ ℕ0
33		oveq2 6087	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑛C𝑘) = (𝑛C𝑚))
34	33	eleq1d 2307	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝑛C𝑘) ∈ ℕ0 ↔ (𝑛C𝑚) ∈ ℕ0))
35	34	cbvralv 2786	. . . 4 ⊢ (∀𝑘 ∈ ℤ (𝑛C𝑘) ∈ ℕ0 ↔ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0)
36		bcpasc 11187	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
37	36	adantlr 481	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) = ((𝑛 + 1)C𝑘))
38		oveq2 6087	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → (𝑛C𝑚) = (𝑛C𝑘))
39	38	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C𝑘) ∈ ℕ0))
40	39	rspccva 2928	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑛C𝑘) ∈ ℕ0)
41		peano2zm 9665	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈ ℤ)
42		oveq2 6087	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (𝑛C𝑚) = (𝑛C(𝑘 − 1)))
43	42	eleq1d 2307	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((𝑛C𝑚) ∈ ℕ0 ↔ (𝑛C(𝑘 − 1)) ∈ ℕ0))
44	43	rspccva 2928	. . . . . . . . . 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 9607	. . . . . . . 8 ⊢ ((∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
47	46	adantll 480	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛C𝑘) + (𝑛C(𝑘 − 1))) ∈ ℕ0)
48	37, 47	eqeltrrd 2316	. . . . . 6 ⊢ (((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℤ (𝑛C𝑚) ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑛 + 1)C𝑘) ∈ ℕ0)
49	48	ralrimiva 2623	. . . . 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 9743	. 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑘 ∈ ℤ (𝑁C𝑘) ∈ ℕ0)
53		oveq2 6087	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑁C𝑘) = (𝑁C𝐾))
54	53	eleq1d 2307	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑁C𝑘) ∈ ℕ0 ↔ (𝑁C𝐾) ∈ ℕ0))
55	54	rspccva 2928	. 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 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  (class class class)co 6079  0cc0 8173  1c1 8174   + caddc 8176   − cmin 8491  ℕ₀cn0 9546  ℤcz 9627  ...cfz 10394  Ccbc 11168

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-seqfrec 10868  df-fac 11147  df-bc 11169

This theorem is referenced by:  bccl2  11189  bcn2m1  11191  bcn2p1  11192  binomlem  12233  bcxmas  12239