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Theorem bccmpl 11338
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 11334 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
2 fznn0sub2 10841 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N
) )
3 bcval2 11334 . . . . . 6  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
42, 3syl 15 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) ) ) )
5 elfznn0 10838 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
6 faccl 11314 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
75, 6syl 15 . . . . . . . . . 10  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
87nncnd 9778 . . . . . . . . 9  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
92, 8syl 15 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
10 elfznn0 10838 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
11 faccl 11314 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
1210, 11syl 15 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
1312nncnd 9778 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
149, 13mulcomd 8872 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 K )  x.  ( ! `  ( N  -  K )
) ) )
15 elfz3nn0 10839 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
16 elfzelz 10814 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
17 nn0cn 9991 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
18 zcn 10045 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
19 nncan 9092 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( N  -  K )
)  =  K )
2017, 18, 19syl2an 463 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  ( N  -  K )
)  =  K )
2115, 16, 20syl2anc 642 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  =  K )
2221fveq2d 5545 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  ( N  -  K ) ) )  =  ( ! `  K ) )
2322oveq1d 5889 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) )  =  ( ( ! `  K )  x.  ( ! `  ( N  -  K ) ) ) )
2414, 23eqtr4d 2331 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 ( N  -  ( N  -  K
) ) )  x.  ( ! `  ( N  -  K )
) ) )
2524oveq2d 5890 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  ( N  -  K )
) )  x.  ( ! `  ( N  -  K ) ) ) ) )
264, 25eqtr4d 2331 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
271, 26eqtr4d 2331 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
2827adantl 452 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
29 bcval3 11335 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
30 simp1 955 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  N  e.  NN0 )
31 nn0z 10062 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
32 zsubcl 10077 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
3331, 32sylan 457 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
34333adant3 975 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  ZZ )
35 fznn0sub2 10841 . . . . . . . 8  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  e.  ( 0 ... N
) )
3620eleq1d 2362 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  ( N  -  K
) )  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... N ) ) )
3735, 36syl5ib 210 . . . . . . 7  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  K )  e.  ( 0 ... N )  ->  K  e.  ( 0 ... N ) ) )
3837con3d 125 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( -.  K  e.  ( 0 ... N
)  ->  -.  ( N  -  K )  e.  ( 0 ... N
) ) )
39383impia 1148 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  -.  ( N  -  K )  e.  ( 0 ... N ) )
40 bcval3 11335 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  -  K
)  e.  ZZ  /\  -.  ( N  -  K
)  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4130, 34, 39, 40syl3anc 1182 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4229, 41eqtr4d 2331 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
43423expa 1151 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  -.  K  e.  ( 0 ... N
) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
4428, 43pm2.61dan 766 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798   !cfa 11304    _C cbc 11331
This theorem is referenced by:  bcnn  11340  bcnp1n  11342  bcp1m1  11348  basellem3  20336  chtublem  20466  bcmax  20533  bcp1ctr  20534  bcnm1  24111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332
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