Theorem bcn2 10503

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 8874	. . 3 |- 2 e. NN
2		bcval5 10502	. . 3 |- ( ( N e. NN0 /\ 2 e. NN ) -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
3	1, 2	mpan2 421	. 2 |- ( N e. NN0 -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
4		2m1e1 8831	. . . . . . . 8 |- ( 2 - 1 ) = 1
5	4	oveq2i 5778	. . . . . . 7 |- ( ( N - 2 ) + ( 2 - 1 ) ) = ( ( N - 2 ) + 1 )
6		nn0cn 8980	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
7		2cn 8784	. . . . . . . . 9 |- 2 e. CC
8		ax-1cn 7706	. . . . . . . . 9 |- 1 e. CC
9		npncan 7976	. . . . . . . . 9 |- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
10	7, 8, 9	mp3an23 1307	. . . . . . . 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	syl5eqr 2184	. . . . . 6 |- ( N e. NN0 -> ( ( N - 2 ) + 1 ) = ( N - 1 ) )
13	12	seqeq1d 10217	. . . . 5 |- ( N e. NN0 -> seq ( ( N - 2 ) + 1 ) ( x. , _I ) = seq ( N - 1 ) ( x. , _I ) )
14	13	fveq1d 5416	. . . 4 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( seq ( N - 1 ) ( x. , _I ) ` N ) )
15		nn0z 9067	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
16		peano2zm 9085	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
17	15, 16	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. ZZ )
18		uzid 9333	. . . . . . . . 9 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
19	15, 18	syl 14	. . . . . . . 8 |- ( N e. NN0 -> N e. ( ZZ>= ` N ) )
20		npcan 7964	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
21	6, 8, 20	sylancl 409	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) + 1 ) = N )
22	21	fveq2d 5418	. . . . . . . 8 |- ( N e. NN0 -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
23	19, 22	eleqtrrd 2217	. . . . . . 7 |- ( N e. NN0 -> N e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
24		eluzelcn 9330	. . . . . . . . 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 5471	. . . . . . . . . 10 |- ( x e. CC -> ( _I ` x ) = x )
27	26	eleq1d 2206	. . . . . . . . 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 7740	. . . . . . . 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 10234	. . . . . 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 10226	. . . . . . . 8 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( _I ` ( N - 1 ) ) )
34		fvi 5471	. . . . . . . . 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 2170	. . . . . . 7 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( N - 1 ) )
37		fvi 5471	. . . . . . 7 |- ( N e. NN0 -> ( _I ` N ) = N )
38	36, 37	oveq12d 5785	. . . . . 6 |- ( N e. NN0 -> ( ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) x. ( _I ` N ) ) = ( ( N - 1 ) x. N ) )
39	32, 38	eqtrd 2170	. . . . 5 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( ( N - 1 ) x. N ) )
40		subcl 7954	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
41	6, 8, 40	sylancl 409	. . . . . 6 |- ( N e. NN0 -> ( N - 1 ) e. CC )
42	41, 6	mulcomd 7780	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) x. N ) = ( N x. ( N - 1 ) ) )
43	39, 42	eqtrd 2170	. . . 4 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
44	14, 43	eqtrd 2170	. . 3 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
45		fac2 10470	. . . 4 |- ( ! ` 2 ) = 2
46	45	a1i 9	. . 3 |- ( N e. NN0 -> ( ! ` 2 ) = 2 )
47	44, 46	oveq12d 5785	. 2 |- ( N e. NN0 -> ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) = ( ( N x. ( N - 1 ) ) / 2 ) )
48	3, 47	eqtrd 2170	1 |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _I cid 4205   ` cfv 5118  (class class class)co 5767   CCcc 7611   1c1 7614   + caddc 7616   x. cmul 7618   - cmin 7926   / cdiv 8425   NNcn 8713   2c2 8764   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   seqcseq 10211   !cfa 10464   _C cbc 10486

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-fz 9784  df-seqfrec 10212  df-fac 10465  df-bc 10487

This theorem is referenced by:  bcp1m1  10504