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Theorem bccmpl 11601
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 11597 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
2 fznn0sub2 11087 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N
) )
3 bcval2 11597 . . . . . 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 11084 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
6 faccl 11577 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
75, 6syl 16 . . . . . . . . . 10  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
87nncnd 10017 . . . . . . . . 9  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
92, 8syl 16 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
10 elfznn0 11084 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
11 faccl 11577 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
1210, 11syl 16 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
1312nncnd 10017 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
149, 13mulcomd 9110 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 K )  x.  ( ! `  ( N  -  K )
) ) )
15 elfz3nn0 11085 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
16 elfzelz 11060 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
17 nn0cn 10232 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
18 zcn 10288 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
19 nncan 9331 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( N  -  K )
)  =  K )
2017, 18, 19syl2an 465 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  ( N  -  K )
)  =  K )
2115, 16, 20syl2anc 644 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  =  K )
2221fveq2d 5733 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  ( N  -  K ) ) )  =  ( ! `  K ) )
2322oveq1d 6097 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) )  =  ( ( ! `  K )  x.  ( ! `  ( N  -  K ) ) ) )
2414, 23eqtr4d 2472 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 ( N  -  ( N  -  K
) ) )  x.  ( ! `  ( N  -  K )
) ) )
2524oveq2d 6098 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  ( N  -  K )
) )  x.  ( ! `  ( N  -  K ) ) ) ) )
264, 25eqtr4d 2472 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
271, 26eqtr4d 2472 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
2827adantl 454 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
29 bcval3 11598 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
30 simp1 958 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  N  e.  NN0 )
31 nn0z 10305 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
32 zsubcl 10320 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
3331, 32sylan 459 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
34333adant3 978 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  ZZ )
35 fznn0sub2 11087 . . . . . . . 8  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  e.  ( 0 ... N
) )
3620eleq1d 2503 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  ( N  -  K
) )  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... N ) ) )
3735, 36syl5ib 212 . . . . . . 7  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  K )  e.  ( 0 ... N )  ->  K  e.  ( 0 ... N ) ) )
3837con3d 128 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( -.  K  e.  ( 0 ... N
)  ->  -.  ( N  -  K )  e.  ( 0 ... N
) ) )
39383impia 1151 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  -.  ( N  -  K )  e.  ( 0 ... N ) )
40 bcval3 11598 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  -  K
)  e.  ZZ  /\  -.  ( N  -  K
)  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4130, 34, 39, 40syl3anc 1185 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4229, 41eqtr4d 2472 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
43423expa 1154 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  -.  K  e.  ( 0 ... N
) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
4428, 43pm2.61dan 768 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   CCcc 8989   0cc0 8991    x. cmul 8996    - cmin 9292    / cdiv 9678   NNcn 10001   NN0cn0 10222   ZZcz 10283   ...cfz 11044   !cfa 11567    _C cbc 11594
This theorem is referenced by:  bcnn  11604  bcnp1n  11606  bcp1m1  11612  basellem3  20866  chtublem  20996  bcmax  21063  bcp1ctr  21064  bcnm1  25202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-seq 11325  df-fac 11568  df-bc 11595
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