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Theorem bccmpl 10158
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 10154 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2 fznn0sub2 9535 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
3 bcval2 10154 . . . . . 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 9524 . . . . . . . . . . 11 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - K ) e. NN0 )
65faccld 10140 . . . . . . . . . 10 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
76nncnd 8434 . . . . . . . . 9 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
82, 7syl 14 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
9 elfznn0 9524 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
109faccld 10140 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
1110nncnd 8434 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
128, 11mulcomd 7507 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
13 elfz3nn0 9525 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> N e. NN0 )
14 elfzelz 9438 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. ZZ )
15 nn0cn 8681 . . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
16 zcn 8753 . . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
17 nncan 7709 . . . . . . . . . . 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 5309 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - ( N - K ) ) ) = ( ! ` K ) )
2120oveq1d 5667 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
2212, 21eqtr4d 2123 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) )
2322oveq2d 5668 . . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
244, 23eqtr4d 2123 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
251, 24eqtr4d 2123 . . 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 10155 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )
28 simp1 943 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> N e. NN0 )
29 nn0z 8768 . . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
30 zsubcl 8789 . . . . . . 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 963 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N - K ) e. ZZ )
33 fznn0sub2 9535 . . . . . . . 8 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - ( N - K ) ) e. ( 0 ... N ) )
3418eleq1d 2156 . . . . . . . 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 596 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( -. K e. ( 0 ... N ) -> -. ( N - K ) e. ( 0 ... N ) ) )
37363impia 1140 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> -. ( N - K ) e. ( 0 ... N ) )
38 bcval3 10155 . . . . 5 |- ( ( N e. NN0 /\ ( N - K ) e. ZZ /\ -. ( N - K ) e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
3928, 32, 37, 38syl3anc 1174 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4027, 39eqtr4d 2123 . . 3 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
41403expa 1143 . 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 8760 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
4429adantr 270 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
45 fzdcel 9452 . . . 4 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
4642, 43, 44, 45syl3anc 1174 . . 3 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
47 exmiddc 782 . . 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 747 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 664  DECID wdc 780   /\ w3a 924   = wceq 1289   e. wcel 1438   ` cfv 5015  (class class class)co 5652   CCcc 7346   0cc0 7348   x. cmul 7353   - cmin 7651   / cdiv 8137   NN0cn0 8671   ZZcz 8748   ...cfz 9422   !cfa 10129   _C cbc 10151
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-fz 9423  df-iseq 9849  df-fac 10130  df-bc 10152
This theorem is referenced by:  bcnn  10161  bcnp1n  10163  bcp1m1  10169  bcnm1  10176
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