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Theorem bcn2 11537
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
StepHypRef Expression
1 2nn 10065 . . 3  |-  2  e.  NN
2 bcval5 11536 . . 3  |-  ( ( N  e.  NN0  /\  2  e.  NN )  ->  ( N  _C  2
)  =  ( (  seq  ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
) )
31, 2mpan2 653 . 2  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( (  seq  (
( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! ` 
2 ) ) )
4 2m1e1 10027 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 6031 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 10163 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 10002 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 8981 . . . . . . . . 9  |-  1  e.  CC
9 npncan 9255 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
107, 8, 9mp3an23 1271 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
116, 10syl 16 . . . . . . 7  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  -  1 ) )
125, 11syl5eqr 2433 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1312seqeq1d 11256 . . . . 5  |-  ( N  e.  NN0  ->  seq  (
( N  -  2 )  +  1 ) (  x.  ,  _I  )  =  seq  ( N  -  1 ) (  x.  ,  _I  )
)
1413fveq1d 5670 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )
)
15 nn0z 10236 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
16 peano2zm 10252 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1715, 16syl 16 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
18 uzid 10432 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
1915, 18syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
20 npcan 9246 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
216, 8, 20sylancl 644 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2221fveq2d 5672 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2319, 22eleqtrrd 2464 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
24 seqm1 11267 . . . . . . 7  |-  ( ( ( N  -  1 )  e.  ZZ  /\  N  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )  -> 
(  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  x.  (  _I  `  N ) ) )
2517, 23, 24syl2anc 643 . . . . . 6  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  x.  (  _I  `  N ) ) )
26 seq1 11263 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  (  _I  `  ( N  -  1
) ) )
2717, 26syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  (  _I  `  ( N  -  1
) ) )
28 fvi 5722 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
2917, 28syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3027, 29eqtrd 2419 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  ( N  - 
1 ) )
31 fvi 5722 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3230, 31oveq12d 6038 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq  ( N  - 
1 ) (  x.  ,  _I  ) `  ( N  -  1
) )  x.  (  _I  `  N ) )  =  ( ( N  -  1 )  x.  N ) )
3325, 32eqtrd 2419 . . . . 5  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( ( N  -  1 )  x.  N ) )
34 subcl 9237 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
356, 8, 34sylancl 644 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  CC )
3635, 6mulcomd 9042 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
3733, 36eqtrd 2419 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
3814, 37eqtrd 2419 . . 3  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
39 fac2 11499 . . . 4  |-  ( ! `
 2 )  =  2
4039a1i 11 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4138, 40oveq12d 6038 . 2  |-  ( N  e.  NN0  ->  ( (  seq  ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
)  =  ( ( N  x.  ( N  -  1 ) )  /  2 ) )
423, 41eqtrd 2419 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    _I cid 4434   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420    seq cseq 11250   !cfa 11493    _C cbc 11520
This theorem is referenced by:  bcp1m1  11538  bpoly3  25818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-seq 11251  df-fac 11494  df-bc 11521
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