Theorem bcn2 11089

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 9364	. . 3 |- 2 e. NN
2		bcval5 11088	. . 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 9320	. . . . . . . 8 |- ( 2 - 1 ) = 1
5	4	oveq2i 6039	. . . . . . 7 |- ( ( N - 2 ) + ( 2 - 1 ) ) = ( ( N - 2 ) + 1 )
6		nn0cn 9471	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
7		2cn 9273	. . . . . . . . 9 |- 2 e. CC
8		ax-1cn 8185	. . . . . . . . 9 |- 1 e. CC
9		npncan 8459	. . . . . . . . 9 |- ( ( N e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( N - 2 ) + ( 2 - 1 ) ) = ( N - 1 ) )
10	7, 8, 9	mp3an23 1366	. . . . . . . 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 2278	. . . . . 6 |- ( N e. NN0 -> ( ( N - 2 ) + 1 ) = ( N - 1 ) )
13	12	seqeq1d 10778	. . . . 5 |- ( N e. NN0 -> seq ( ( N - 2 ) + 1 ) ( x. , _I ) = seq ( N - 1 ) ( x. , _I ) )
14	13	fveq1d 5650	. . . 4 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( seq ( N - 1 ) ( x. , _I ) ` N ) )
15		nn0z 9560	. . . . . . . 8 |- ( N e. NN0 -> N e. ZZ )
16		peano2zm 9578	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
17	15, 16	syl 14	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. ZZ )
18		uzid 9831	. . . . . . . . 9 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
19	15, 18	syl 14	. . . . . . . 8 |- ( N e. NN0 -> N e. ( ZZ>= ` N ) )
20		npcan 8447	. . . . . . . . . 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 5652	. . . . . . . 8 |- ( N e. NN0 -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
23	19, 22	eleqtrrd 2311	. . . . . . 7 |- ( N e. NN0 -> N e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
24		eluzelcn 9828	. . . . . . . . 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 5712	. . . . . . . . . 10 |- ( x e. CC -> ( _I ` x ) = x )
27	26	eleq1d 2300	. . . . . . . . 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 8219	. . . . . . . 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 10798	. . . . . 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 10787	. . . . . . . 8 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( _I ` ( N - 1 ) ) )
34		fvi 5712	. . . . . . . . 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 2264	. . . . . . 7 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) = ( N - 1 ) )
37		fvi 5712	. . . . . . 7 |- ( N e. NN0 -> ( _I ` N ) = N )
38	36, 37	oveq12d 6046	. . . . . 6 |- ( N e. NN0 -> ( ( seq ( N - 1 ) ( x. , _I ) ` ( N - 1 ) ) x. ( _I ` N ) ) = ( ( N - 1 ) x. N ) )
39	32, 38	eqtrd 2264	. . . . 5 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( ( N - 1 ) x. N ) )
40		subcl 8437	. . . . . . 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 8260	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) x. N ) = ( N x. ( N - 1 ) ) )
43	39, 42	eqtrd 2264	. . . 4 |- ( N e. NN0 -> ( seq ( N - 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
44	14, 43	eqtrd 2264	. . 3 |- ( N e. NN0 -> ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) = ( N x. ( N - 1 ) ) )
45		fac2 11056	. . . 4 |- ( ! ` 2 ) = 2
46	45	a1i 9	. . 3 |- ( N e. NN0 -> ( ! ` 2 ) = 2 )
47	44, 46	oveq12d 6046	. 2 |- ( N e. NN0 -> ( ( seq ( ( N - 2 ) + 1 ) ( x. , _I ) ` N ) / ( ! ` 2 ) ) = ( ( N x. ( N - 1 ) ) / 2 ) )
48	3, 47	eqtrd 2264	1 |- ( N e. NN0 -> ( N _C 2 ) = ( ( N x. ( N - 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _I cid 4391   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   x. cmul 8097   - cmin 8409   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   seqcseq 10772   !cfa 11050   _C cbc 11072

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-fz 10306  df-seqfrec 10773  df-fac 11051  df-bc 11073

This theorem is referenced by:  bcp1m1  11090