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Theorem bcp1nk 9786
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 9120 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  0  e.  ZZ )
2 elfzel2 9119 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
3 elfzelz 9121 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
4 1zzd 8459 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  1  e.  ZZ )
5 fzaddel 9153 . . . . . 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 1171 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( K  e.  ( 0 ... N )  <->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) ) )
76ibi 174 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) )
8 1e0p1 8599 . . . . 5  |-  1  =  ( 0  +  1 )
98oveq1i 5553 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  =  ( ( 0  +  1 ) ... ( N  +  1 ) )
107, 9syl6eleqr 2173 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
11 bcm1k 9784 . . 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 14 . 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 8551 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
14 ax-1cn 7131 . . . . . . 7  |-  1  e.  CC
15 pncan 7381 . . . . . . 7  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
1613, 14, 15sylancl 404 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( K  +  1 )  -  1 )  =  K )
1716oveq2d 5559 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  _C  K ) )
18 bcp1n 9785 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
1917, 18eqtrd 2114 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2016oveq2d 5559 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  K ) )
2120oveq1d 5558 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  (
( K  +  1 )  -  1 ) )  /  ( K  +  1 ) )  =  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) )
2219, 21oveq12d 5561 . . 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 9777 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
2423rpcnd 8856 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  CC )
252peano2zd 8553 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  ZZ )
2625zred 8550 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  RR )
273zred 8550 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
282zred 8550 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
29 elfzle2 9123 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
3028ltp1d 8075 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  <  ( N  +  1 ) )
3127, 28, 26, 29, 30lelttrd 7301 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  <  ( N  +  1 ) )
32 znnsub 8483 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  ( N  +  1
)  e.  ZZ )  ->  ( K  < 
( N  +  1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
333, 25, 32syl2anc 403 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( K  <  ( N  + 
1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
3431, 33mpbid 145 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
3526, 34nndivred 8155 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  RR )
3635recnd 7209 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  CC )
3734nnred 8119 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  RR )
38 elfznn0 9207 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
39 nn0p1nn 8394 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
4038, 39syl 14 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  NN )
4137, 40nndivred 8155 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  RR )
4241recnd 7209 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  CC )
4324, 36, 42mulassd 7204 . . . 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 8551 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
4534nncnd 8120 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
4640nncnd 8120 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  CC )
4734nnap0d 8151 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K ) #  0 )
4840nnap0d 8151 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 ) #  0 )
4944, 45, 46, 47, 48dmdcanap2d 7974 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5049oveq2d 5559 . . . 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 2114 . . 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 2114 . 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 2114 1  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048    < clt 7215    - cmin 7346    / cdiv 7827   NNcn 8106   NN0cn0 8355   ZZcz 8432   ...cfz 9105    _C cbc 9771
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-q 8786  df-rp 8816  df-fz 9106  df-iseq 9522  df-fac 9750  df-bc 9772
This theorem is referenced by: (None)
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