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Theorem bccmpl 9681
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 9677 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2 fznn0sub2 9139 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
3 bcval2 9677 . . . . . 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 9130 . . . . . . . . . . 11 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - K ) e. NN0 )
65faccld 9663 . . . . . . . . . 10 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
76nncnd 8053 . . . . . . . . 9 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
82, 7syl 14 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
9 elfznn0 9130 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
109faccld 9663 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
1110nncnd 8053 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
128, 11mulcomd 7140 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
13 elfz3nn0 9131 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> N e. NN0 )
14 elfzelz 9045 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. ZZ )
15 nn0cn 8298 . . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
16 zcn 8356 . . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17 nncan 7337 . . . . . . . . . . 11 |- ( ( N e. CC /\ K e. CC ) -> ( N - ( N - K ) ) = K )
1815, 16, 17syl2an 283 . . . . . . . . . 10 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - ( N - K ) ) = K )
1913, 14, 18syl2anc 403 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( N - ( N - K ) ) = K )
2019fveq2d 5202 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - ( N - K ) ) ) = ( ! ` K ) )
2120oveq1d 5547 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
2212, 21eqtr4d 2116 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) )
2322oveq2d 5548 . . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
244, 23eqtr4d 2116 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
251, 24eqtr4d 2116 . . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( N _C ( N - K ) ) )
2625adantl 271 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
27 bcval3 9678 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )
28 simp1 938 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> N e. NN0 )
29 nn0z 8371 . . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
30 zsubcl 8392 . . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N - K ) e. ZZ )
3129, 30sylan 277 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - K ) e. ZZ )
32313adant3 958 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N - K ) e. ZZ )
33 fznn0sub2 9139 . . . . . . . 8 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - ( N - K ) ) e. ( 0 ... N ) )
3418eleq1d 2147 . . . . . . . 8 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - ( N - K ) ) e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
3533, 34syl5ib 152 . . . . . . 7 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - K ) e. ( 0 ... N ) -> K e. ( 0 ... N ) ) )
3635con3d 593 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( -. K e. ( 0 ... N ) -> -. ( N - K ) e. ( 0 ... N ) ) )
37363impia 1135 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> -. ( N - K ) e. ( 0 ... N ) )
38 bcval3 9678 . . . . 5 |- ( ( N e. NN0 /\ ( N - K ) e. ZZ /\ -. ( N - K ) e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
3928, 32, 37, 38syl3anc 1169 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4027, 39eqtr4d 2116 . . 3 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
41403expa 1138 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
42 simpr 108 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> K e. ZZ )
43 0zd 8363 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
4429adantr 270 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
45 fzdcel 9059 . . . 4 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
4642, 43, 44, 45syl3anc 1169 . . 3 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
47 exmiddc 777 . . 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 744 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 102   \/ wo 661  DECID wdc 775   /\ w3a 919   = wceq 1284   e. wcel 1433   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   x. cmul 6986   - cmin 7279   / cdiv 7760   NN0cn0 8288   ZZcz 8351   ...cfz 9029   !cfa 9652   _C cbc 9674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-fz 9030  df-iseq 9432  df-fac 9653  df-bc 9675
This theorem is referenced by:  bcnn  9684  bcnp1n  9686  bcp1m1  9692  bcnm1  9699
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