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Theorem bccmpl 9622
Description: "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
Assertion
Ref Expression
bccmpl  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K
)  =  ( N  _C  ( N  -  K ) ) )

Proof of Theorem bccmpl
StepHypRef Expression
1 bcval2 9618 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
2 fznn0sub2 9087 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N
) )
3 bcval2 9618 . . . . . 6  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
42, 3syl 14 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
5 elfznn0 9077 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
65faccld 9604 . . . . . . . . . 10  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
76nncnd 8004 . . . . . . . . 9  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
82, 7syl 14 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
9 elfznn0 9077 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
109faccld 9604 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
1110nncnd 8004 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
128, 11mulcomd 7106 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 K )  x.  ( ! `  ( N  -  K )
) ) )
13 elfz3nn0 9078 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
14 elfzelz 8992 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
15 nn0cn 8249 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
16 zcn 8307 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
17 nncan 7303 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( N  -  K )
)  =  K )
1815, 16, 17syl2an 277 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  ( N  -  K )
)  =  K )
1913, 14, 18syl2anc 397 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  =  K )
2019fveq2d 5210 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  ( N  -  K ) ) )  =  ( ! `  K ) )
2120oveq1d 5555 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) )  =  ( ( ! `  K )  x.  ( ! `  ( N  -  K ) ) ) )
2212, 21eqtr4d 2091 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 ( N  -  ( N  -  K
) ) )  x.  ( ! `  ( N  -  K )
) ) )
2322oveq2d 5556 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  ( N  -  K )
) )  x.  ( ! `  ( N  -  K ) ) ) ) )
244, 23eqtr4d 2091 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
251, 24eqtr4d 2091 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
2625adantl 266 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
27 bcval3 9619 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
28 simp1 915 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  N  e.  NN0 )
29 nn0z 8322 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 zsubcl 8343 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
3129, 30sylan 271 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
32313adant3 935 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  ZZ )
33 fznn0sub2 9087 . . . . . . . 8  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  e.  ( 0 ... N
) )
3418eleq1d 2122 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  ( N  -  K
) )  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... N ) ) )
3533, 34syl5ib 147 . . . . . . 7  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  K )  e.  ( 0 ... N )  ->  K  e.  ( 0 ... N ) ) )
3635con3d 571 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( -.  K  e.  ( 0 ... N
)  ->  -.  ( N  -  K )  e.  ( 0 ... N
) ) )
37363impia 1112 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  -.  ( N  -  K )  e.  ( 0 ... N ) )
38 bcval3 9619 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  -  K
)  e.  ZZ  /\  -.  ( N  -  K
)  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
3928, 32, 37, 38syl3anc 1146 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4027, 39eqtr4d 2091 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
41403expa 1115 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  -.  K  e.  ( 0 ... N
) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
42 simpr 107 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  K  e.  ZZ )
43 0zd 8314 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  0  e.  ZZ )
4429adantr 265 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  N  e.  ZZ )
45 fzdcel 9006 . . . 4  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  (
0 ... N ) )
4642, 43, 44, 45syl3anc 1146 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  -> DECID  K  e.  ( 0 ... N ) )
47 exmiddc 755 . . 3  |-  (DECID  K  e.  ( 0 ... N
)  ->  ( K  e.  ( 0 ... N
)  \/  -.  K  e.  ( 0 ... N
) ) )
4846, 47syl 14 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( K  e.  ( 0 ... N )  \/  -.  K  e.  ( 0 ... N
) ) )
4926, 41, 48mpjaodan 722 1  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K
)  =  ( N  _C  ( N  -  K ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639  DECID wdc 753    /\ w3a 896    = wceq 1259    e. wcel 1409   ` cfv 4930  (class class class)co 5540   CCcc 6945   0cc0 6947    x. cmul 6952    - cmin 7245    / cdiv 7725   NN0cn0 8239   ZZcz 8302   ...cfz 8976   !cfa 9593    _C cbc 9615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-q 8652  df-fz 8977  df-iseq 9376  df-fac 9594  df-bc 9616
This theorem is referenced by:  bcnn  9625  bcnp1n  9627  bcp1m1  9633  bcnm1  9640
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