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Theorem bcp1nk 12363
Description: The proportion of one binomial coefficient to another with  N and  K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
bcp1nk  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )

Proof of Theorem bcp1nk
StepHypRef Expression
1 elfzel1 11687 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  0  e.  ZZ )
2 elfzel2 11686 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
3 elfzelz 11688 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
4 1zzd 10895 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  1  e.  ZZ )
5 fzaddel 11718 . . . . . 6  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( K  e.  ( 0 ... N )  <-> 
( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  + 
1 ) ) ) )
61, 2, 3, 4, 5syl22anc 1229 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( K  e.  ( 0 ... N )  <->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) ) )
76ibi 241 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) )
8 1e0p1 11004 . . . . 5  |-  1  =  ( 0  +  1 )
98oveq1i 6294 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  =  ( ( 0  +  1 ) ... ( N  +  1 ) )
107, 9syl6eleqr 2566 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
11 bcm1k 12361 . . 3  |-  ( ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( ( N  +  1 )  _C  ( ( K  +  1 )  - 
1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  + 
1 )  -  1 ) )  /  ( K  +  1 ) ) ) )
1210, 11syl 16 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( ( N  +  1 )  _C  ( ( K  +  1 )  - 
1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  + 
1 )  -  1 ) )  /  ( K  +  1 ) ) ) )
133zcnd 10967 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
14 ax-1cn 9550 . . . . . . 7  |-  1  e.  CC
15 pncan 9826 . . . . . . 7  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
1613, 14, 15sylancl 662 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( K  +  1 )  -  1 )  =  K )
1716oveq2d 6300 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  _C  K ) )
18 bcp1n 12362 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
1917, 18eqtrd 2508 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2016oveq2d 6300 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  K ) )
2120oveq1d 6299 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  (
( K  +  1 )  -  1 ) )  /  ( K  +  1 ) )  =  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) )
2219, 21oveq12d 6302 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  _C  (
( K  +  1 )  -  1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  /  ( K  + 
1 ) ) )  =  ( ( ( N  _C  K )  x.  ( ( N  +  1 )  / 
( ( N  + 
1 )  -  K
) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  +  1 ) ) ) )
23 bcrpcl 12354 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
2423rpcnd 11258 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  CC )
252peano2zd 10969 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  ZZ )
2625zred 10966 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  RR )
273zred 10966 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
282zred 10966 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
29 elfzle2 11690 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
3028ltp1d 10476 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  <  ( N  +  1 ) )
3127, 28, 26, 29, 30lelttrd 9739 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  <  ( N  +  1 ) )
32 znnsub 10909 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  ( N  +  1
)  e.  ZZ )  ->  ( K  < 
( N  +  1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
333, 25, 32syl2anc 661 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( K  <  ( N  + 
1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
3431, 33mpbid 210 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
3526, 34nndivred 10584 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  RR )
3635recnd 9622 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  CC )
3734nnred 10551 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  RR )
38 elfznn0 11770 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
39 nn0p1nn 10835 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
4038, 39syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  NN )
4137, 40nndivred 10584 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  RR )
4241recnd 9622 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  CC )
4324, 36, 42mulassd 9619 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  _C  K )  x.  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( ( N  +  1 )  / 
( ( N  + 
1 )  -  K
) )  x.  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) ) ) ) )
4425zcnd 10967 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
4534nncnd 10552 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
4640nncnd 10552 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  CC )
4734nnne0d 10580 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4840nnne0d 10580 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  =/=  0 )
4944, 45, 46, 47, 48dmdcan2d 10350 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5049oveq2d 6300 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  _C  K
)  x.  ( ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5143, 50eqtrd 2508 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  _C  K )  x.  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5222, 51eqtrd 2508 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  _C  (
( K  +  1 )  -  1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5312, 52eqtrd 2508 1  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   class class class wbr 4447  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    - cmin 9805    / cdiv 10206   NNcn 10536   NN0cn0 10795   ZZcz 10864   ...cfz 11672    _C cbc 12348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-seq 12076  df-fac 12322  df-bc 12349
This theorem is referenced by:  sylow1lem1  16424
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