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Theorem bccmpl 10628
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 10624 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
2 fznn0sub2 10027 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N
) )
3 bcval2 10624 . . . . . 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 10016 . . . . . . . . . . 11  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
65faccld 10610 . . . . . . . . . 10  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
76nncnd 8847 . . . . . . . . 9  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
82, 7syl 14 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
9 elfznn0 10016 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
109faccld 10610 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
1110nncnd 8847 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
128, 11mulcomd 7899 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 K )  x.  ( ! `  ( N  -  K )
) ) )
13 elfz3nn0 10017 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
14 elfzelz 9928 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
15 nn0cn 9100 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
16 zcn 9172 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  K  e.  CC )
17 nncan 8104 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( N  -  K )
)  =  K )
1815, 16, 17syl2an 287 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  ( N  -  K )
)  =  K )
1913, 14, 18syl2anc 409 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  =  K )
2019fveq2d 5472 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  ( N  -  K ) ) )  =  ( ! `  K ) )
2120oveq1d 5839 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  ( N  -  K ) ) )  x.  ( ! `  ( N  -  K
) ) )  =  ( ( ! `  K )  x.  ( ! `  ( N  -  K ) ) ) )
2212, 21eqtr4d 2193 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =  ( ( ! `
 ( N  -  ( N  -  K
) ) )  x.  ( ! `  ( N  -  K )
) ) )
2322oveq2d 5840 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  ( N  -  K )
) )  x.  ( ! `  ( N  -  K ) ) ) ) )
244, 23eqtr4d 2193 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  ( N  -  K ) )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
251, 24eqtr4d 2193 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
2625adantl 275 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
27 bcval3 10625 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
28 simp1 982 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  N  e.  NN0 )
29 nn0z 9187 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 zsubcl 9208 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
3129, 30sylan 281 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  -  K
)  e.  ZZ )
32313adant3 1002 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  ZZ )
33 fznn0sub2 10027 . . . . . . . 8  |-  ( ( N  -  K )  e.  ( 0 ... N )  ->  ( N  -  ( N  -  K ) )  e.  ( 0 ... N
) )
3418eleq1d 2226 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  ( N  -  K
) )  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... N ) ) )
3533, 34syl5ib 153 . . . . . . 7  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  -  K )  e.  ( 0 ... N )  ->  K  e.  ( 0 ... N ) ) )
3635con3d 621 . . . . . 6  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( -.  K  e.  ( 0 ... N
)  ->  -.  ( N  -  K )  e.  ( 0 ... N
) ) )
37363impia 1182 . . . . 5  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  -.  ( N  -  K )  e.  ( 0 ... N ) )
38 bcval3 10625 . . . . 5  |-  ( ( N  e.  NN0  /\  ( N  -  K
)  e.  ZZ  /\  -.  ( N  -  K
)  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
3928, 32, 37, 38syl3anc 1220 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  ( N  -  K
) )  =  0 )
4027, 39eqtr4d 2193 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
41403expa 1185 . 2  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  -.  K  e.  ( 0 ... N
) )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K )
) )
42 simpr 109 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  K  e.  ZZ )
43 0zd 9179 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  0  e.  ZZ )
4429adantr 274 . . . 4  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  N  e.  ZZ )
45 fzdcel 9942 . . . 4  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  (
0 ... N ) )
4642, 43, 44, 45syl3anc 1220 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  -> DECID  K  e.  ( 0 ... N ) )
47 exmiddc 822 . . 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 788 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 103    \/ wo 698  DECID wdc 820    /\ w3a 963    = wceq 1335    e. wcel 2128   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732    x. cmul 7737    - cmin 8046    / cdiv 8545   NN0cn0 9090   ZZcz 9167   ...cfz 9912   !cfa 10599    _C cbc 10621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-fz 9913  df-seqfrec 10345  df-fac 10600  df-bc 10622
This theorem is referenced by:  bcnn  10631  bcnp1n  10633  bcp1m1  10639  bcnm1  10646
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