Theorem bcn2 10677

Description: Binomial coefficient: N choose 2. (Contributed by Mario Carneiro, 22-May-2014.)

Assertion

Ref	Expression
bcn2	|- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Proof of Theorem bcn2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn 9018	. . 3 |- 2 e. NN
2		bcval5 10676	. . 3 |- ( ( N e. NN0 /\ 2 e. NN ) -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
3	1, 2	mpan2 422	. 2 |- ( N e. NN0 -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
4		2m1e1 8975	. . . . . . . 8 |- ( 2 - 1 ) = 1
5	4	oveq2i 5853	. . . . . . 7 |- ( ( N - 2 ) + ( 2 - 1 ) ) = ( ( N - 2 ) + 1 )
6		nn0cn 9124	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
7		2cn 8928	. . . . . . . . 9 |- 2 e. CC
8		ax-1cn 7846	. . . . . . . . 9 |- 1 e. CC
9		npncan 8119	. . . . . . . . 9 |- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
10	7, 8, 9	mp3an23 1319	. . . . . . . 8 |- ( N e. CC -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
11	6, 10	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
12	5, 11	eqtr3id 2213	. . . . . 6 |- ( N e. NN0 -> ( ( N - 2 ) + 1 ) = ( N - 1 ) )
13	12	seqeq1d 10386	. . . . 5 |- ( N e. NN0 -> seq ( ( N - 2 ) + 1 ) ( x. , _I ) = seq ( N - 1 ) ( x. , _I ) )
14	13	fveq1d 5488	. . . 4 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( seq ( N - 1 ) ( x. , _I ) ` N ) )
15		nn0z 9211	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
16		peano2zm 9229	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
17	15, 16	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. ZZ )
18		uzid 9480	. . . . . . . . 9 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
19	15, 18	syl 14	. . . . . . . 8 |- ( N e. NN0 -> N e. ( ZZ>= ` N ) )
20		npcan 8107	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
21	6, 8, 20	sylancl 410	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) + 1 ) = N )
22	21	fveq2d 5490	. . . . . . . 8 |- ( N e. NN0 -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
23	19, 22	eleqtrrd 2246	. . . . . . 7 |- ( N e. NN0 -> N e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
24		eluzelcn 9477	. . . . . . . . 9 |- ( x e. ( ZZ>= ` ( N - 1 ) ) -> x e. CC )
25	24	adantl 275	. . . . . . . 8 |- ( ( N e. NN0 /\ x e. ( ZZ>= ` ( N - 1 ) ) ) -> x e. CC )
26		fvi 5543	. . . . . . . . . 10 |- ( x e. CC -> ( _I ` x ) = x )
27	26	eleq1d 2235	. . . . . . . . 9 |- ( x e. CC -> ( ( _I ` x ) e. CC <-> x e. CC ) )
28	27	ibir 176	. . . . . . . 8 |- ( x e. CC -> ( _I ` x ) e. CC )
29	25, 28	syl 14	. . . . . . 7 |- ( ( N e. NN0 /\ x e. ( ZZ>= ` ( N - 1 ) ) ) -> ( _I ` x ) e. CC )
30		mulcl 7880	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
31	30	adantl 275	. . . . . . 7 |- ( ( N e. NN0 /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
32	17, 23, 29, 31	seq3m1 10403	. . . . . 6 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) x. ( _I ` N ) ) )
33	17, 29, 31	seq3-1 10395	. . . . . . . 8 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( _I ` ( N - 1 ) ) )
34		fvi 5543	. . . . . . . . 9 |- ( ( N - 1 ) e. ZZ -> ( _I ` ( N - 1 ) ) = ( N - 1 ) )
35	17, 34	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( _I ` ( N - 1 ) ) = ( N - 1 ) )
36	33, 35	eqtrd 2198	. . . . . . 7 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( N - 1 ) )
37		fvi 5543	. . . . . . 7 |- ( N e. NN0 -> ( _I ` N ) = N )
38	36, 37	oveq12d 5860	. . . . . 6 |- ( N e. NN0 -> ( ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) x. ( _I ` N ) ) = ( ( N - 1 ) x. N ) )
39	32, 38	eqtrd 2198	. . . . 5 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( ( N - 1 ) x. N ) )
40		subcl 8097	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
41	6, 8, 40	sylancl 410	. . . . . 6 |- ( N e. NN0 -> ( N - 1 ) e. CC )
42	41, 6	mulcomd 7920	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) x. N ) = ( N x. ( N - 1 ) ) )
43	39, 42	eqtrd 2198	. . . 4 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
44	14, 43	eqtrd 2198	. . 3 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
45		fac2 10644	. . . 4 |- ( ! ` 2 ) = 2
46	45	a1i 9	. . 3 |- ( N e. NN0 -> ( ! ` 2 ) = 2 )
47	44, 46	oveq12d 5860	. 2 |- ( N e. NN0 -> ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) = ( ( N x. ( N - 1 ) ) / 2 ) )
48	3, 47	eqtrd 2198	1 |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _I cid 4266   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   / cdiv 8568   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   seqcseq 10380   !cfa 10638   _C cbc 10660

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-fz 9945  df-seqfrec 10381  df-fac 10639  df-bc 10661

This theorem is referenced by:  bcp1m1  10678