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Theorem bcp1nk 11345
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 10813 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  0  e.  ZZ )
2 elfzel2 10812 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
3 elfzelz 10814 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
4 1z 10069 . . . . . . 7  |-  1  e.  ZZ
54a1i 10 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  1  e.  ZZ )
6 fzaddel 10842 . . . . . 6  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( K  e.  ( 0 ... N )  <-> 
( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  + 
1 ) ) ) )
71, 2, 3, 5, 6syl22anc 1183 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( K  e.  ( 0 ... N )  <->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) ) )
87ibi 232 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) )
9 1e0p1 10168 . . . . 5  |-  1  =  ( 0  +  1 )
109oveq1i 5884 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  =  ( ( 0  +  1 ) ... ( N  +  1 ) )
118, 10syl6eleqr 2387 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
12 bcm1k 11343 . . 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 ) ) ) )
1311, 12syl 15 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( ( N  +  1 )  _C  ( ( K  +  1 )  - 
1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  + 
1 )  -  1 ) )  /  ( K  +  1 ) ) ) )
143zcnd 10134 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
15 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
16 pncan 9073 . . . . . . 7  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
1714, 15, 16sylancl 643 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( K  +  1 )  -  1 )  =  K )
1817oveq2d 5890 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  _C  K ) )
19 bcp1n 11344 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2018, 19eqtrd 2328 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2117oveq2d 5890 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  K ) )
2221oveq1d 5889 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  (
( K  +  1 )  -  1 ) )  /  ( K  +  1 ) )  =  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) )
2320, 22oveq12d 5892 . . 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 ) ) ) )
24 bcrpcl 11337 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
2524rpcnd 10408 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  CC )
262peano2zd 10136 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  ZZ )
2726zred 10133 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  RR )
283zred 10133 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
292zred 10133 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
30 elfzle2 10816 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
3129ltp1d 9703 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  <  ( N  +  1 ) )
3228, 29, 27, 30, 31lelttrd 8990 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  <  ( N  +  1 ) )
33 znnsub 10080 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  ( N  +  1
)  e.  ZZ )  ->  ( K  < 
( N  +  1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
343, 26, 33syl2anc 642 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( K  <  ( N  + 
1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
3532, 34mpbid 201 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
3627, 35nndivred 9810 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  RR )
3736recnd 8877 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  CC )
3835nnred 9777 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  RR )
39 elfznn0 10838 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
40 nn0p1nn 10019 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
4139, 40syl 15 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  NN )
4238, 41nndivred 9810 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  RR )
4342recnd 8877 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  CC )
4425, 37, 43mulassd 8874 . . . 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 ) ) ) ) )
4537, 43mulcomd 8872 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
4626zcnd 10134 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
4735nncnd 9778 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
4841nncnd 9778 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  CC )
4935nnne0d 9806 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
5041nnne0d 9806 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  =/=  0 )
5146, 47, 48, 49, 50dmdcand 9581 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ( N  +  1 )  -  K )  /  ( K  +  1 ) )  x.  ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5245, 51eqtrd 2328 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5352oveq2d 5890 . . . 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 ) ) ) )
5444, 53eqtrd 2328 . . 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 ) ) ) )
5523, 54eqtrd 2328 . 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 ) ) ) )
5613, 55eqtrd 2328 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 176    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798    _C cbc 11331
This theorem is referenced by:  sylow1lem1  14925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332
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