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Theorem bccmpl 11555
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 11551 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
2 fznn0sub2 11042 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N
) )
3 bcval2 11551 . . . . . 6  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
42, 3syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
5 elfznn0 11039 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
6 faccl 11531 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
75, 6syl 16 . . . . . . . . . 10  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
87nncnd 9972 . . . . . . . . 9  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
92, 8syl 16 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
10 elfznn0 11039 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
11 faccl 11531 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
1210, 11syl 16 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
1312nncnd 9972 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
149, 13mulcomd 9065 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 K )  x.  ( ! `  ( N  -  K )
) ) )
15 elfz3nn0 11040 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
16 elfzelz 11015 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
17 nn0cn 10187 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
18 zcn 10243 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
19 nncan 9286 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( N  -  K )
)  =  K )
2017, 18, 19syl2an 464 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  ( N  -  K )
)  =  K )
2115, 16, 20syl2anc 643 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  =  K )
2221fveq2d 5691 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  ( N  -  K ) ) )  =  ( ! `  K ) )
2322oveq1d 6055 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) )  =  ( ( ! `  K )  x.  ( ! `  ( N  -  K ) ) ) )
2414, 23eqtr4d 2439 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 ( N  -  ( N  -  K
) ) )  x.  ( ! `  ( N  -  K )
) ) )
2524oveq2d 6056 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  ( N  -  K )
) )  x.  ( ! `  ( N  -  K ) ) ) ) )
264, 25eqtr4d 2439 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
271, 26eqtr4d 2439 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
2827adantl 453 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
29 bcval3 11552 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
30 simp1 957 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  N  e.  NN0 )
31 nn0z 10260 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
32 zsubcl 10275 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
3331, 32sylan 458 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
34333adant3 977 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  ZZ )
35 fznn0sub2 11042 . . . . . . . 8  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  e.  ( 0 ... N
) )
3620eleq1d 2470 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  ( N  -  K
) )  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... N ) ) )
3735, 36syl5ib 211 . . . . . . 7  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  K )  e.  ( 0 ... N )  ->  K  e.  ( 0 ... N ) ) )
3837con3d 127 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( -.  K  e.  ( 0 ... N
)  ->  -.  ( N  -  K )  e.  ( 0 ... N
) ) )
39383impia 1150 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  -.  ( N  -  K )  e.  ( 0 ... N ) )
40 bcval3 11552 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  -  K
)  e.  ZZ  /\  -.  ( N  -  K
)  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4130, 34, 39, 40syl3anc 1184 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4229, 41eqtr4d 2439 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
43423expa 1153 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  -.  K  e.  ( 0 ... N
) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
4428, 43pm2.61dan 767 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ...cfz 10999   !cfa 11521    _C cbc 11548
This theorem is referenced by:  bcnn  11558  bcnp1n  11560  bcp1m1  11566  basellem3  20818  chtublem  20948  bcmax  21015  bcp1ctr  21016  bcnm1  25154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-fac 11522  df-bc 11549
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