ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bccmpl Unicode version
Theorem bccmpl 10846
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 10842 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2 fznn0sub2 10203 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
3 bcval2 10842 . . . . . 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 10189 . . . . . . . . . . 11 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - K ) e. NN0 )
65faccld 10828 . . . . . . . . . 10 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
76nncnd 9004 . . . . . . . . 9 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
82, 7syl 14 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
9 elfznn0 10189 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
109faccld 10828 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
1110nncnd 9004 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
128, 11mulcomd 8048 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
13 elfz3nn0 10190 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> N e. NN0 )
14 elfzelz 10100 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. ZZ )
15 nn0cn 9259 . . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
16 zcn 9331 . . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17 nncan 8255 . . . . . . . . . . 11 |- ( ( N e. CC /\ K e. CC ) -> ( N - ( N - K ) ) = K )
1815, 16, 17syl2an 289 . . . . . . . . . 10 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - ( N - K ) ) = K )
1913, 14, 18syl2anc 411 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( N - ( N - K ) ) = K )
2019fveq2d 5562 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - ( N - K ) ) ) = ( ! ` K ) )
2120oveq1d 5937 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
2212, 21eqtr4d 2232 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) )
2322oveq2d 5938 . . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
244, 23eqtr4d 2232 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
251, 24eqtr4d 2232 . . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( N _C ( N - K ) ) )
2625adantl 277 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
27 bcval3 10843 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )
28 simp1 999 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> N e. NN0 )
29 nn0z 9346 . . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
30 zsubcl 9367 . . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N - K ) e. ZZ )
3129, 30sylan 283 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - K ) e. ZZ )
32313adant3 1019 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N - K ) e. ZZ )
33 fznn0sub2 10203 . . . . . . . 8 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - ( N - K ) ) e. ( 0 ... N ) )
3418eleq1d 2265 . . . . . . . 8 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - ( N - K ) ) e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
3533, 34imbitrid 154 . . . . . . 7 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - K ) e. ( 0 ... N ) -> K e. ( 0 ... N ) ) )
3635con3d 632 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( -. K e. ( 0 ... N ) -> -. ( N - K ) e. ( 0 ... N ) ) )
37363impia 1202 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> -. ( N - K ) e. ( 0 ... N ) )
38 bcval3 10843 . . . . 5 |- ( ( N e. NN0 /\ ( N - K ) e. ZZ /\ -. ( N - K ) e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
3928, 32, 37, 38syl3anc 1249 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4027, 39eqtr4d 2232 . . 3 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
41403expa 1205 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
42 simpr 110 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> K e. ZZ )
43 0zd 9338 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
4429adantr 276 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
45 fzdcel 10115 . . . 4 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
4642, 43, 44, 45syl3anc 1249 . . 3 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
47 exmiddc 837 . . 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 799 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 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   x. cmul 7884   - cmin 8197   / cdiv 8699   NN0cn0 9249   ZZcz 9326   ...cfz 10083   !cfa 10817   _C cbc 10839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-fz 10084  df-seqfrec 10540  df-fac 10818  df-bc 10840
This theorem is referenced by:  bcnn  10849  bcnp1n  10851  bcp1m1  10857  bcnm1  10864
  Copyright terms: Public domain W3C validator