Theorem bcn2 10746

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 9082	. . 3 |- 2 e. NN
2		bcval5 10745	. . 3 |- ( ( N e. NN0 /\ 2 e. NN ) -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
3	1, 2	mpan2 425	. 2 |- ( N e. NN0 -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) )
4		2m1e1 9039	. . . . . . . 8 |- ( 2 - 1 ) = 1
5	4	oveq2i 5888	. . . . . . 7 |- ( ( N - 2 ) + ( 2 - 1 ) ) = ( ( N - 2 ) + 1 )
6		nn0cn 9188	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
7		2cn 8992	. . . . . . . . 9 |- 2 e. CC
8		ax-1cn 7906	. . . . . . . . 9 |- 1 e. CC
9		npncan 8180	. . . . . . . . 9 |- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
10	7, 8, 9	mp3an23 1329	. . . . . . . 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 2224	. . . . . 6 |- ( N e. NN0 -> ( ( N - 2 ) + 1 ) = ( N - 1 ) )
13	12	seqeq1d 10453	. . . . 5 |- ( N e. NN0 -> seq ( ( N - 2 ) + 1 ) ( x. , _I ) = seq ( N - 1 ) ( x. , _I ) )
14	13	fveq1d 5519	. . . 4 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( seq ( N - 1 ) ( x. , _I ) ` N ) )
15		nn0z 9275	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
16		peano2zm 9293	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
17	15, 16	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. ZZ )
18		uzid 9544	. . . . . . . . 9 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
19	15, 18	syl 14	. . . . . . . 8 |- ( N e. NN0 -> N e. ( ZZ>= ` N ) )
20		npcan 8168	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
21	6, 8, 20	sylancl 413	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) + 1 ) = N )
22	21	fveq2d 5521	. . . . . . . 8 |- ( N e. NN0 -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
23	19, 22	eleqtrrd 2257	. . . . . . 7 |- ( N e. NN0 -> N e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
24		eluzelcn 9541	. . . . . . . . 9 |- ( x e. ( ZZ>= ` ( N - 1 ) ) -> x e. CC )
25	24	adantl 277	. . . . . . . 8 |- ( ( N e. NN0 /\ x e. ( ZZ>= ` ( N - 1 ) ) ) -> x e. CC )
26		fvi 5575	. . . . . . . . . 10 |- ( x e. CC -> ( _I ` x ) = x )
27	26	eleq1d 2246	. . . . . . . . 9 |- ( x e. CC -> ( ( _I ` x ) e. CC <-> x e. CC ) )
28	27	ibir 177	. . . . . . . 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 7940	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
31	30	adantl 277	. . . . . . 7 |- ( ( N e. NN0 /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
32	17, 23, 29, 31	seq3m1 10470	. . . . . 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 10462	. . . . . . . 8 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( _I ` ( N - 1 ) ) )
34		fvi 5575	. . . . . . . . 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 2210	. . . . . . 7 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( N - 1 ) )
37		fvi 5575	. . . . . . 7 |- ( N e. NN0 -> ( _I ` N ) = N )
38	36, 37	oveq12d 5895	. . . . . 6 |- ( N e. NN0 -> ( ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) x. ( _I ` N ) ) = ( ( N - 1 ) x. N ) )
39	32, 38	eqtrd 2210	. . . . 5 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( ( N - 1 ) x. N ) )
40		subcl 8158	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
41	6, 8, 40	sylancl 413	. . . . . 6 |- ( N e. NN0 -> ( N - 1 ) e. CC )
42	41, 6	mulcomd 7981	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) x. N ) = ( N x. ( N - 1 ) ) )
43	39, 42	eqtrd 2210	. . . 4 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
44	14, 43	eqtrd 2210	. . 3 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
45		fac2 10713	. . . 4 |- ( ! ` 2 ) = 2
46	45	a1i 9	. . 3 |- ( N e. NN0 -> ( ! ` 2 ) = 2 )
47	44, 46	oveq12d 5895	. 2 |- ( N e. NN0 -> ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) = ( ( N x. ( N - 1 ) ) / 2 ) )
48	3, 47	eqtrd 2210	1 |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _I cid 4290   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   + caddc 7816   x. cmul 7818   - cmin 8130   / cdiv 8631   NNcn 8921   2c2 8972   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   seqcseq 10447   !cfa 10707   _C cbc 10729

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-fz 10011  df-seqfrec 10448  df-fac 10708  df-bc 10730

This theorem is referenced by:  bcp1m1  10747