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Theorem bccmpl 10963
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 10959 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2 fznn0sub2 10312 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
3 bcval2 10959 . . . . . 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 10298 . . . . . . . . . . 11 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - K ) e. NN0 )
65faccld 10945 . . . . . . . . . 10 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
76nncnd 9112 . . . . . . . . 9 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
82, 7syl 14 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
9 elfznn0 10298 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
109faccld 10945 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
1110nncnd 9112 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
128, 11mulcomd 8156 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
13 elfz3nn0 10299 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> N e. NN0 )
14 elfzelz 10209 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. ZZ )
15 nn0cn 9367 . . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
16 zcn 9439 . . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17 nncan 8363 . . . . . . . . . . 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 5627 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - ( N - K ) ) ) = ( ! ` K ) )
2120oveq1d 6009 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
2212, 21eqtr4d 2265 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) )
2322oveq2d 6010 . . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
244, 23eqtr4d 2265 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
251, 24eqtr4d 2265 . . 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 10960 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )
28 simp1 1021 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> N e. NN0 )
29 nn0z 9454 . . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
30 zsubcl 9475 . . . . . . 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 1041 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N - K ) e. ZZ )
33 fznn0sub2 10312 . . . . . . . 8 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - ( N - K ) ) e. ( 0 ... N ) )
3418eleq1d 2298 . . . . . . . 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 634 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( -. K e. ( 0 ... N ) -> -. ( N - K ) e. ( 0 ... N ) ) )
37363impia 1224 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> -. ( N - K ) e. ( 0 ... N ) )
38 bcval3 10960 . . . . 5 |- ( ( N e. NN0 /\ ( N - K ) e. ZZ /\ -. ( N - K ) e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
3928, 32, 37, 38syl3anc 1271 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4027, 39eqtr4d 2265 . . 3 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
41403expa 1227 . 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 9446 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
4429adantr 276 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
45 fzdcel 10224 . . . 4 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
4642, 43, 44, 45syl3anc 1271 . . 3 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
47 exmiddc 841 . . 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 803 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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5314  (class class class)co 5994   CCcc 7985   0cc0 7987   x. cmul 7992   - cmin 8305   / cdiv 8807   NN0cn0 9357   ZZcz 9434   ...cfz 10192   !cfa 10934   _C cbc 10956
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-fz 10193  df-seqfrec 10657  df-fac 10935  df-bc 10957
This theorem is referenced by:  bcnn  10966  bcnp1n  10968  bcp1m1  10974  bcnm1  10981
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