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Theorem bcn2 12116
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 10500 . . 3  |-  2  e.  NN
2 bcval5 12115 . . 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 10457 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 6123 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 10610 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 10413 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 9361 . . . . . . . . 9  |-  1  e.  CC
9 npncan 9651 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
107, 8, 9mp3an23 1306 . . . . . . . 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 2489 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1312seqeq1d 11833 . . . . 5  |-  ( N  e.  NN0  ->  seq (
( N  -  2 )  +  1 ) (  x.  ,  _I  )  =  seq ( N  -  1 ) (  x.  ,  _I  ) )
1413fveq1d 5714 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )
)
15 nn0z 10690 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
16 peano2zm 10709 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1715, 16syl 16 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
18 uzid 10896 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
1915, 18syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
20 npcan 9640 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
216, 8, 20sylancl 662 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2221fveq2d 5716 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2319, 22eleqtrrd 2520 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
24 seqm1 11844 . . . . . . 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 11840 . . . . . . . . 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 5769 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
2917, 28syl 16 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3027, 29eqtrd 2475 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  ( N  -  1 ) )  =  ( N  - 
1 ) )
31 fvi 5769 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3230, 31oveq12d 6130 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq ( N  - 
1 ) (  x.  ,  _I  ) `  ( N  -  1
) )  x.  (  _I  `  N ) )  =  ( ( N  -  1 )  x.  N ) )
3325, 32eqtrd 2475 . . . . 5  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( ( N  -  1 )  x.  N ) )
34 subcl 9630 . . . . . . 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 9428 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
3733, 36eqtrd 2475 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
3814, 37eqtrd 2475 . . 3  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ) `  N )  =  ( N  x.  ( N  -  1
) ) )
39 fac2 12078 . . . 4  |-  ( ! `
 2 )  =  2
4039a1i 11 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4138, 40oveq12d 6130 . 2  |-  ( N  e.  NN0  ->  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ) `  N )  /  ( ! `  2 )
)  =  ( ( N  x.  ( N  -  1 ) )  /  2 ) )
423, 41eqtrd 2475 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    _I cid 4652   ` cfv 5439  (class class class)co 6112   CCcc 9301   1c1 9304    + caddc 9306    x. cmul 9308    - cmin 9616    / cdiv 10014   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882    seqcseq 11827   !cfa 12072    _C cbc 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-seq 11828  df-fac 12073  df-bc 12100
This theorem is referenced by:  bcp1m1  12117  bpoly3  28223
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