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Theorem bccmpl 13096
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 13092 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2 fznn0sub2 12446 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )
3 bcval2 13092 . . . . . 6 |- ( ( N - K ) e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
42, 3syl 17 . . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
5 elfznn0 12433 . . . . . . . . . . 11 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - K ) e. NN0 )
6 faccl 13070 . . . . . . . . . . 11 |- ( ( N - K ) e. NN0 -> ( ! ` ( N - K ) ) e. NN )
75, 6syl 17 . . . . . . . . . 10 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
87nncnd 11036 . . . . . . . . 9 |- ( ( N - K ) e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
92, 8syl 17 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
10 elfznn0 12433 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
11 faccl 13070 . . . . . . . . . 10 |- ( K e. NN0 -> ( ! ` K ) e. NN )
1210, 11syl 17 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
1312nncnd 11036 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
149, 13mulcomd 10061 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
15 elfz3nn0 12434 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> N e. NN0 )
16 elfzelz 12342 . . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. ZZ )
17 nn0cn 11302 . . . . . . . . . . 11 |- ( N e. NN0 -> N e. CC )
18 zcn 11382 . . . . . . . . . . 11 |- ( K e. ZZ -> K e. CC )
19 nncan 10310 . . . . . . . . . . 11 |- ( ( N e. CC /\ K e. CC ) -> ( N - ( N - K ) ) = K )
2017, 18, 19syl2an 494 . . . . . . . . . 10 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - ( N - K ) ) = K )
2115, 16, 20syl2anc 693 . . . . . . . . 9 |- ( K e. ( 0 ... N ) -> ( N - ( N - K ) ) = K )
2221fveq2d 6195 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - ( N - K ) ) ) = ( ! ` K ) )
2322oveq1d 6665 . . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) = ( ( ! ` K ) x. ( ! ` ( N - K ) ) ) )
2414, 23eqtr4d 2659 . . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) = ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) )
2524oveq2d 6666 . . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - ( N - K ) ) ) x. ( ! ` ( N - K ) ) ) ) )
264, 25eqtr4d 2659 . . . 4 |- ( K e. ( 0 ... N ) -> ( N _C ( N - K ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
271, 26eqtr4d 2659 . . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( N _C ( N - K ) ) )
2827adantl 482 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
29 bcval3 13093 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = 0 )
30 simp1 1061 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> N e. NN0 )
31 nn0z 11400 . . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
32 zsubcl 11419 . . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N - K ) e. ZZ )
3331, 32sylan 488 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N - K ) e. ZZ )
34333adant3 1081 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N - K ) e. ZZ )
35 fznn0sub2 12446 . . . . . . . 8 |- ( ( N - K ) e. ( 0 ... N ) -> ( N - ( N - K ) ) e. ( 0 ... N ) )
3620eleq1d 2686 . . . . . . . 8 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - ( N - K ) ) e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
3735, 36syl5ib 234 . . . . . . 7 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( ( N - K ) e. ( 0 ... N ) -> K e. ( 0 ... N ) ) )
3837con3d 148 . . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( -. K e. ( 0 ... N ) -> -. ( N - K ) e. ( 0 ... N ) ) )
39383impia 1261 . . . . 5 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> -. ( N - K ) e. ( 0 ... N ) )
40 bcval3 13093 . . . . 5 |- ( ( N e. NN0 /\ ( N - K ) e. ZZ /\ -. ( N - K ) e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4130, 34, 39, 40syl3anc 1326 . . . 4 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C ( N - K ) ) = 0 )
4229, 41eqtr4d 2659 . . 3 |- ( ( N e. NN0 /\ K e. ZZ /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
43423expa 1265 . 2 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ -. K e. ( 0 ... N ) ) -> ( N _C K ) = ( N _C ( N - K ) ) )
4428, 43pm2.61dan 832 1 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = ( N _C ( N - K ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   !cfa 13060   _C cbc 13089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090
This theorem is referenced by:  bcnn  13099  bcnp1n  13101  bcp1m1  13107  bcnm1  13114  basellem3  24809  chtublem  24936  bcmax  25003  bcp1ctr  25004
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