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Theorem bcp1nk 12371
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 11693 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  0  e.  ZZ )
2 elfzel2 11692 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
3 elfzelz 11694 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
4 1zzd 10898 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  1  e.  ZZ )
5 fzaddel 11724 . . . . . 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 1228 . . . . 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 11009 . . . . 5  |-  1  =  ( 0  +  1 )
98oveq1i 6288 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  =  ( ( 0  +  1 ) ... ( N  +  1 ) )
107, 9syl6eleqr 2540 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
11 bcm1k 12369 . . 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 10972 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
14 ax-1cn 9550 . . . . . . 7  |-  1  e.  CC
15 pncan 9828 . . . . . . 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 6294 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  _C  K ) )
18 bcp1n 12370 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
1917, 18eqtrd 2482 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2016oveq2d 6294 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  K ) )
2120oveq1d 6293 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  (
( K  +  1 )  -  1 ) )  /  ( K  +  1 ) )  =  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) )
2219, 21oveq12d 6296 . . 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 12362 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
2423rpcnd 11264 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  CC )
252peano2zd 10974 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  ZZ )
2625zred 10971 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  RR )
273zred 10971 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
282zred 10971 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
29 elfzle2 11696 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
3028ltp1d 10479 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  <  ( N  +  1 ) )
3127, 28, 26, 29, 30lelttrd 9740 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  <  ( N  +  1 ) )
32 znnsub 10913 . . . . . . . . 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 10587 . . . . . 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 10554 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  RR )
38 elfznn0 11776 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
39 nn0p1nn 10838 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
4038, 39syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  NN )
4137, 40nndivred 10587 . . . . . 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 10972 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
4534nncnd 10555 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
4640nncnd 10555 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  CC )
4734nnne0d 10583 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4840nnne0d 10583 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  =/=  0 )
4944, 45, 46, 47, 48dmdcan2d 10353 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5049oveq2d 6294 . . . 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 2482 . . 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 2482 . 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 2482 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 1381    e. wcel 1802   class class class wbr 4434  (class class class)co 6278   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    - cmin 9807    / cdiv 10209   NNcn 10539   NN0cn0 10798   ZZcz 10867   ...cfz 11678    _C cbc 12356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  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 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-n0 10799  df-z 10868  df-uz 11088  df-rp 11227  df-fz 11679  df-seq 12084  df-fac 12330  df-bc 12357
This theorem is referenced by:  sylow1lem1  16489
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