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Theorem bcn2 10136
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 8547 . . 3  |-  2  e.  NN
2 ibcval5 10135 . . 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 8510 . . . . . . . 8  |-  ( 2  -  1 )  =  1
54oveq2i 5645 . . . . . . 7  |-  ( ( N  -  2 )  +  ( 2  -  1 ) )  =  ( ( N  - 
2 )  +  1 )
6 nn0cn 8653 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
7 2cn 8464 . . . . . . . . 9  |-  2  e.  CC
8 ax-1cn 7417 . . . . . . . . 9  |-  1  e.  CC
9 npncan 7682 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( N  -  2 )  +  ( 2  -  1 ) )  =  ( N  - 
1 ) )
107, 8, 9mp3an23 1265 . . . . . . . 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 2134 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
13 iseqeq1 9822 . . . . . 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 5291 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  (  seq ( N  - 
1 ) (  x.  ,  _I  ,  CC ) `  N )
)
16 nn0z 8740 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
17 peano2zm 8758 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
1816, 17syl 14 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  -  1 )  e.  ZZ )
19 uzid 9002 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ( ZZ>= `  N )
)
2016, 19syl 14 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  N )
)
21 npcan 7670 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
226, 8, 21sylancl 404 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  +  1 )  =  N )
2322fveq2d 5293 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
2420, 23eleqtrrd 2167 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
25 eluzelcn 8999 . . . . . . . . 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 5345 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (  _I  `  x )  =  x )
2827eleq1d 2156 . . . . . . . . 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 7448 . . . . . . . 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 9852 . . . . . 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 9838 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1
) )  =  (  _I  `  ( N  -  1 ) ) )
35 fvi 5345 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  ZZ  ->  (  _I  `  ( N  - 
1 ) )  =  ( N  -  1 ) )
3618, 35syl 14 . . . . . . . 8  |-  ( N  e.  NN0  ->  (  _I 
`  ( N  - 
1 ) )  =  ( N  -  1 ) )
3734, 36eqtrd 2120 . . . . . . 7  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1
) )  =  ( N  -  1 ) )
38 fvi 5345 . . . . . . 7  |-  ( N  e.  NN0  ->  (  _I 
`  N )  =  N )
3937, 38oveq12d 5652 . . . . . 6  |-  ( N  e.  NN0  ->  ( (  seq ( N  - 
1 ) (  x.  ,  _I  ,  CC ) `  ( N  -  1 ) )  x.  (  _I  `  N ) )  =  ( ( N  - 
1 )  x.  N
) )
4033, 39eqtrd 2120 . . . . 5  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( ( N  -  1 )  x.  N ) )
41 subcl 7660 . . . . . . 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 7488 . . . . 5  |-  ( N  e.  NN0  ->  ( ( N  -  1 )  x.  N )  =  ( N  x.  ( N  -  1 ) ) )
4440, 43eqtrd 2120 . . . 4  |-  ( N  e.  NN0  ->  (  seq ( N  -  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( N  x.  ( N  -  1 ) ) )
4515, 44eqtrd 2120 . . 3  |-  ( N  e.  NN0  ->  (  seq ( ( N  - 
2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  =  ( N  x.  ( N  -  1 ) ) )
46 fac2 10103 . . . 4  |-  ( ! `
 2 )  =  2
4746a1i 9 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 2 )  =  2 )
4845, 47oveq12d 5652 . 2  |-  ( N  e.  NN0  ->  ( (  seq ( ( N  -  2 )  +  1 ) (  x.  ,  _I  ,  CC ) `  N )  /  ( ! ` 
2 ) )  =  ( ( N  x.  ( N  -  1
) )  /  2
) )
493, 48eqtrd 2120 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 1289    e. wcel 1438    _I cid 4106   ` cfv 5002  (class class class)co 5634   CCcc 7327   1c1 7330    + caddc 7332    x. cmul 7334    - cmin 7632    / cdiv 8113   NNcn 8394   2c2 8444   NN0cn0 8643   ZZcz 8720   ZZ>=cuz 8988    seqcseq4 9816   !cfa 10097    _C cbc 10119
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-fz 9394  df-iseq 9818  df-fac 10098  df-bc 10120
This theorem is referenced by:  bcp1m1  10137
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