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Theorem bcn2 9788
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2nn 8260 . . 3  |-  2  e.  NN
2 ibcval5 9787 . . 3  |-  ( ( N  e.  NN0  /\  2  e.  NN )  ->  ( N  _C  2
)  =  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  /  ( ! ` 
2 ) ) )
31, 2mpan2 416 . 2  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( (  seq (
( N  -  2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N
)  /  ( ! `
 2 ) ) )
4 2m1e1 8223 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 5554 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 8365 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 8177 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 7131 . . . . . . . . 9  |-  1  e.  CC
9 npncan 7396 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
107, 8, 9mp3an23 1261 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
116, 10syl 14 . . . . . . 7  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  -  1 ) )
125, 11syl5eqr 2128 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
13 iseqeq1 9524 . . . . . 6  |-  ( ( ( N  -  2 )  +  1 )  =  ( N  - 
1 )  ->  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ,  CC )  =  seq ( N  - 
1 ) (  x.  ,  _I  ,  CC ) )
1412, 13syl 14 . . . . 5  |-  ( N  e.  NN0  ->  seq (
( N  -  2 )  +  1 ) (  x.  ,  _I  ,  CC )  =  seq ( N  -  1
) (  x.  ,  _I  ,  CC ) )
1514fveq1d 5211 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  (  seq ( N  - 
1 ) (  x.  ,  _I  ,  CC ) `  N )
)
16 nn0z 8452 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
17 peano2zm 8470 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1816, 17syl 14 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
19 uzid 8714 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
2016, 19syl 14 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
21 npcan 7384 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
226, 8, 21sylancl 404 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2322fveq2d 5213 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2420, 23eleqtrrd 2159 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
25 eluzelcn 8711 . . . . . . . . 9  |-  ( x  e.  ( ZZ>= `  ( N  -  1 ) )  ->  x  e.  CC )
2625adantl 271 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  x  e.  ( ZZ>= `  ( N  -  1
) ) )  ->  x  e.  CC )
27 fvi 5262 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (  _I  `  x )  =  x )
2827eleq1d 2148 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
(  _I  `  x
)  e.  CC  <->  x  e.  CC ) )
2928ibir 175 . . . . . . . 8  |-  ( x  e.  CC  ->  (  _I  `  x )  e.  CC )
3026, 29syl 14 . . . . . . 7  |-  ( ( N  e.  NN0  /\  x  e.  ( ZZ>= `  ( N  -  1
) ) )  -> 
(  _I  `  x
)  e.  CC )
31 mulcl 7162 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
3231adantl 271 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( x  e.  CC  /\  y  e.  CC ) )  ->  ( x  x.  y )  e.  CC )
3318, 24, 30, 32iseqm1 9543 . . . . . 6  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1 ) )  x.  (  _I  `  N ) ) )
3418, 30, 32iseq1 9533 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1
) )  =  (  _I  `  ( N  -  1 ) ) )
35 fvi 5262 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
3618, 35syl 14 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3734, 36eqtrd 2114 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1
) )  =  ( N  -  1 ) )
38 fvi 5262 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3937, 38oveq12d 5561 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq ( N  - 
1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1 ) )  x.  (  _I  `  N ) )  =  ( ( N  - 
1 )  x.  N
) )
4033, 39eqtrd 2114 . . . . 5  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( ( N  -  1 )  x.  N ) )
41 subcl 7374 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
426, 8, 41sylancl 404 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  CC )
4342, 6mulcomd 7202 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
4440, 43eqtrd 2114 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( N  x.  ( N  -  1 ) ) )
4515, 44eqtrd 2114 . . 3  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( N  x.  ( N  -  1 ) ) )
46 fac2 9755 . . . 4  |-  ( ! `
 2 )  =  2
4746a1i 9 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4845, 47oveq12d 5561 . 2  |-  ( N  e.  NN0  ->  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  /  ( ! ` 
2 ) )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
493, 48eqtrd 2114 1  |-  ( N  e.  NN0  ->  ( N  _C  2 )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    _I cid 4051   ` cfv 4932  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046    x. cmul 7048    - cmin 7346    / cdiv 7827   NNcn 8106   2c2 8156   NN0cn0 8355   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521   !cfa 9749    _C cbc 9771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-uz 8701  df-q 8786  df-fz 9106  df-iseq 9522  df-fac 9750  df-bc 9772
This theorem is referenced by:  bcp1m1  9789
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