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Theorem bcn2 12352
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 10682 . . 3  |-  2  e.  NN
2 bcval5 12351 . . 3  |-  ( ( N  e.  NN0  /\  2  e.  NN )  ->  ( N  _C  2
)  =  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
) )
31, 2mpan2 671 . 2  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( (  seq (
( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! ` 
2 ) ) )
4 2m1e1 10639 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 6286 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 10794 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 10595 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 9539 . . . . . . . . 9  |-  1  e.  CC
9 npncan 9829 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
107, 8, 9mp3an23 1311 . . . . . . . 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 2515 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1312seqeq1d 12069 . . . . 5  |-  ( N  e.  NN0  ->  seq (
( N  -  2 )  +  1 ) (  x.  ,  _I  )  =  seq ( N  -  1 ) (  x.  ,  _I  ) )
1413fveq1d 5859 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )
)
15 nn0z 10876 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
16 peano2zm 10895 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1715, 16syl 16 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
18 uzid 11085 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
1915, 18syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
20 npcan 9818 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
216, 8, 20sylancl 662 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2221fveq2d 5861 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2319, 22eleqtrrd 2551 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
24 seqm1 12080 . . . . . . 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 661 . . . . . 6  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( (  seq ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  x.  (  _I  `  N ) ) )
26 seq1 12076 . . . . . . . . 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 5915 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
2917, 28syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3027, 29eqtrd 2501 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  ( N  - 
1 ) )
31 fvi 5915 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3230, 31oveq12d 6293 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq ( N  - 
1 ) (  x.  ,  _I  ) `  ( N  -  1
) )  x.  (  _I  `  N ) )  =  ( ( N  -  1 )  x.  N ) )
3325, 32eqtrd 2501 . . . . 5  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( ( N  -  1 )  x.  N ) )
34 subcl 9808 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
356, 8, 34sylancl 662 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  CC )
3635, 6mulcomd 9606 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
3733, 36eqtrd 2501 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
3814, 37eqtrd 2501 . . 3  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
39 fac2 12314 . . . 4  |-  ( ! `
 2 )  =  2
4039a1i 11 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4138, 40oveq12d 6293 . 2  |-  ( N  e.  NN0  ->  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
)  =  ( ( N  x.  ( N  -  1 ) )  /  2 ) )
423, 41eqtrd 2501 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    _I cid 4783   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794    / cdiv 10195   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071    seqcseq 12063   !cfa 12308    _C cbc 12335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-seq 12064  df-fac 12309  df-bc 12336
This theorem is referenced by:  bcp1m1  12353  bpoly3  29383
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