MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bcn2 Unicode version

Theorem bcn2 11347
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 9893 . . 3  |-  2  e.  NN
2 bcval5 11346 . . 3  |-  ( ( N  e.  NN0  /\  2  e.  NN )  ->  ( N  _C  2
)  =  ( (  seq  ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
) )
31, 2mpan2 652 . 2  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( (  seq  (
( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! ` 
2 ) ) )
4 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
5 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
6 1p1e2 9856 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
74, 5, 5, 6subaddrii 9151 . . . . . . . 8  |-  ( 2  -  1 )  =  1
87oveq2i 5885 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
9 nn0cn 9991 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
10 npncan 9085 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
114, 5, 10mp3an23 1269 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
129, 11syl 15 . . . . . . 7  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  -  1 ) )
138, 12syl5eqr 2342 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1413seqeq1d 11068 . . . . 5  |-  ( N  e.  NN0  ->  seq  (
( N  -  2 )  +  1 ) (  x.  ,  _I  )  =  seq  ( N  -  1 ) (  x.  ,  _I  )
)
1514fveq1d 5543 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )
)
16 nn0z 10062 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
17 peano2zm 10078 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1816, 17syl 15 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
19 uzid 10258 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
2016, 19syl 15 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
21 npcan 9076 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
229, 5, 21sylancl 643 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2322fveq2d 5545 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2420, 23eleqtrrd 2373 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
25 seqm1 11079 . . . . . . 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 ) ) )
2618, 24, 25syl2anc 642 . . . . . 6  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  x.  (  _I  `  N ) ) )
27 seq1 11075 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  (  _I  `  ( N  -  1
) ) )
2818, 27syl 15 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  (  _I  `  ( N  -  1
) ) )
29 fvi 5595 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
3018, 29syl 15 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3128, 30eqtrd 2328 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  ( N  - 
1 ) )
32 fvi 5595 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3331, 32oveq12d 5892 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq  ( N  - 
1 ) (  x.  ,  _I  ) `  ( N  -  1
) )  x.  (  _I  `  N ) )  =  ( ( N  -  1 )  x.  N ) )
3426, 33eqtrd 2328 . . . . 5  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( ( N  -  1 )  x.  N ) )
35 subcl 9067 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
369, 5, 35sylancl 643 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  CC )
3736, 9mulcomd 8872 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
3834, 37eqtrd 2328 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
3915, 38eqtrd 2328 . . 3  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
40 fac2 11310 . . . 4  |-  ( ! `
 2 )  =  2
4140a1i 10 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4239, 41oveq12d 5892 . 2  |-  ( N  e.  NN0  ->  ( (  seq  ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
)  =  ( ( N  x.  ( N  -  1 ) )  /  2 ) )
433, 42eqtrd 2328 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    _I cid 4320   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   !cfa 11304    _C cbc 11331
This theorem is referenced by:  bcp1m1  11348  bpoly3  24865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332
  Copyright terms: Public domain W3C validator