Theorem bcn2 9691

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 8193	. . 3 |- 2 e. NN
2		ibcval5 9690	. . 3 |- ( ( N e. NN0 /\ 2 e. NN ) -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) ` N ) / ( ! ` 2 ) ) )
3	1, 2	mpan2 415	. 2 |- ( N e. NN0 -> ( N _C 2 ) = ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) ` N ) / ( ! ` 2 ) ) )
4		2m1e1 8156	. . . . . . . 8 |- ( 2 - 1 ) = 1
5	4	oveq2i 5543	. . . . . . 7 |- ( ( N - 2 ) + ( 2 - 1 ) ) = ( ( N - 2 ) + 1 )
6		nn0cn 8298	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
7		2cn 8110	. . . . . . . . 9 |- 2 e. CC
8		ax-1cn 7069	. . . . . . . . 9 |- 1 e. CC
9		npncan 7329	. . . . . . . . 9 |- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
10	7, 8, 9	mp3an23 1260	. . . . . . . 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 2127	. . . . . 6 |- ( N e. NN0 -> ( ( N - 2 ) + 1 ) = ( N - 1 ) )
13		iseqeq1 9434	. . . . . 6 |- ( ( ( N - 2 ) + 1 ) = ( N - 1 ) -> seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) = seq ( N - 1 ) ( x. , _I , CC ) )
14	12, 13	syl 14	. . . . 5 |- ( N e. NN0 -> seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) = seq ( N - 1 ) ( x. , _I , CC ) )
15	14	fveq1d 5200	. . . 4 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) ` N ) = ( seq ( N - 1 ) ( x. , _I , CC ) ` N ) )
16		nn0z 8371	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
17		peano2zm 8389	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
18	16, 17	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. ZZ )
19		uzid 8633	. . . . . . . . 9 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
20	16, 19	syl 14	. . . . . . . 8 |- ( N e. NN0 -> N e. ( ZZ>= ` N ) )
21		npcan 7317	. . . . . . . . . 10 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
22	6, 8, 21	sylancl 404	. . . . . . . . 9 |- ( N e. NN0 -> ( ( N - 1 ) + 1 ) = N )
23	22	fveq2d 5202	. . . . . . . 8 |- ( N e. NN0 -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
24	20, 23	eleqtrrd 2158	. . . . . . 7 |- ( N e. NN0 -> N e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
25		cnex 7097	. . . . . . . 8 |- CC e. _V
26	25	a1i 9	. . . . . . 7 |- ( N e. NN0 -> CC e. _V )
27		eluzelcn 8630	. . . . . . . . 9 |- ( x e. ( ZZ>= ` ( N - 1 ) ) -> x e. CC )
28	27	adantl 271	. . . . . . . 8 |- ( ( N e. NN0 /\ x e. ( ZZ>= ` ( N - 1 ) ) ) -> x e. CC )
29		fvi 5251	. . . . . . . . . 10 |- ( x e. CC -> ( _I ` x ) = x )
30	29	eleq1d 2147	. . . . . . . . 9 |- ( x e. CC -> ( ( _I ` x ) e. CC <-> x e. CC ) )
31	30	ibir 175	. . . . . . . 8 |- ( x e. CC -> ( _I ` x ) e. CC )
32	28, 31	syl 14	. . . . . . 7 |- ( ( N e. NN0 /\ x e. ( ZZ>= ` ( N - 1 ) ) ) -> ( _I ` x ) e. CC )
33		mulcl 7100	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
34	33	adantl 271	. . . . . . 7 |- ( ( N e. NN0 /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
35	18, 24, 26, 32, 34	iseqm1 9447	. . . . . 6 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I , CC ) ` N ) = ( ( seq ( N - 1 ) ( x. , _I , CC ) ` ( N - 1 ) ) x. ( _I ` N ) ) )
36	18, 26, 32, 34	iseq1 9442	. . . . . . . 8 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I , CC ) ` ( N - 1 ) ) = ( _I ` ( N - 1 ) ) )
37		fvi 5251	. . . . . . . . 9 |- ( ( N - 1 ) e. ZZ -> ( _I ` ( N - 1 ) ) = ( N - 1 ) )
38	18, 37	syl 14	. . . . . . . 8 |- ( N e. NN0 -> ( _I ` ( N - 1 ) ) = ( N - 1 ) )
39	36, 38	eqtrd 2113	. . . . . . 7 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I , CC ) ` ( N - 1 ) ) = ( N - 1 ) )
40		fvi 5251	. . . . . . 7 |- ( N e. NN0 -> ( _I ` N ) = N )
41	39, 40	oveq12d 5550	. . . . . 6 |- ( N e. NN0 -> ( ( seq ( N - 1 ) ( x. , _I , CC ) ` ( N - 1 ) ) x. ( _I ` N ) ) = ( ( N - 1 ) x. N ) )
42	35, 41	eqtrd 2113	. . . . 5 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I , CC ) ` N ) = ( ( N - 1 ) x. N ) )
43		subcl 7307	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
44	6, 8, 43	sylancl 404	. . . . . 6 |- ( N e. NN0 -> ( N - 1 ) e. CC )
45	44, 6	mulcomd 7140	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) x. N ) = ( N x. ( N - 1 ) ) )
46	42, 45	eqtrd 2113	. . . 4 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I , CC ) ` N ) = ( N x. ( N - 1 ) ) )
47	15, 46	eqtrd 2113	. . 3 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) ` N ) = ( N x. ( N - 1 ) ) )
48		fac2 9658	. . . 4 |- ( ! ` 2 ) = 2
49	48	a1i 9	. . 3 |- ( N e. NN0 -> ( ! ` 2 ) = 2 )
50	47, 49	oveq12d 5550	. 2 |- ( N e. NN0 -> ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I , CC ) ` N ) / ( ! ` 2 ) ) = ( ( N x. ( N - 1 ) ) / 2 ) )
51	3, 50	eqtrd 2113	1 |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   _I cid 4043   ` cfv 4922  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   x. cmul 6986   - cmin 7279   / cdiv 7760   NNcn 8039   2c2 8089   NN0cn0 8288   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431   !cfa 9652   _C cbc 9674

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-fz 9030  df-iseq 9432  df-fac 9653  df-bc 9675

This theorem is referenced by:  bcp1m1  9692