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

Theorem bcn2 11602
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 10125 . . 3  |-  2  e.  NN
2 bcval5 11601 . . 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 10087 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 6084 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 10223 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 10062 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 9040 . . . . . . . . 9  |-  1  e.  CC
9 npncan 9315 . . . . . . . . 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 2481 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1312seqeq1d 11321 . . . . 5  |-  ( N  e.  NN0  ->  seq  (
( N  -  2 )  +  1 ) (  x.  ,  _I  )  =  seq  ( N  -  1 ) (  x.  ,  _I  )
)
1413fveq1d 5722 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )
)
15 nn0z 10296 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
16 peano2zm 10312 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1715, 16syl 16 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
18 uzid 10492 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
1915, 18syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
20 npcan 9306 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
216, 8, 20sylancl 644 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2221fveq2d 5724 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2319, 22eleqtrrd 2512 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
24 seqm1 11332 . . . . . . 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 11328 . . . . . . . . 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 5775 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
2917, 28syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3027, 29eqtrd 2467 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  ( N  - 
1 ) )
31 fvi 5775 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3230, 31oveq12d 6091 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq  ( N  - 
1 ) (  x.  ,  _I  ) `  ( N  -  1
) )  x.  (  _I  `  N ) )  =  ( ( N  -  1 )  x.  N ) )
3325, 32eqtrd 2467 . . . . 5  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( ( N  -  1 )  x.  N ) )
34 subcl 9297 . . . . . . 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 9101 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
3733, 36eqtrd 2467 . . . 4  |-  ( N  e.  NN0  ->  (  seq  ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
3814, 37eqtrd 2467 . . 3  |-  ( N  e.  NN0  ->  (  seq  ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
39 fac2 11564 . . . 4  |-  ( ! `
 2 )  =  2
4039a1i 11 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4138, 40oveq12d 6091 . 2  |-  ( N  e.  NN0  ->  ( (  seq  ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
)  =  ( ( N  x.  ( N  -  1 ) )  /  2 ) )
423, 41eqtrd 2467 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    _I cid 4485   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480    seq cseq 11315   !cfa 11558    _C cbc 11585
This theorem is referenced by:  bcp1m1  11603  bpoly3  26096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-fac 11559  df-bc 11586
  Copyright terms: Public domain W3C validator