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Theorem bcp1n 11344
Description: The proportion of one binomial coefficient to another with  N increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
Assertion
Ref Expression
bcp1n  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )

Proof of Theorem bcp1n
StepHypRef Expression
1 elfz3nn0 10839 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
2 facp1 11309 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 ( N  + 
1 ) )  =  ( ( ! `  N )  x.  ( N  +  1 ) ) )
31, 2syl 15 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
 N )  x.  ( N  +  1 ) ) )
4 fznn0sub 10840 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
5 facp1 11309 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( ( N  -  K )  +  1 ) )  =  ( ( ! `  ( N  -  K
) )  x.  (
( N  -  K
)  +  1 ) ) )
64, 5syl 15 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  -  K )  +  1 ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
71nn0cnd 10036 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  CC )
8 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
98a1i 10 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  1  e.  CC )
10 elfznn0 10838 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
1110nn0cnd 10036 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
127, 9, 11addsubd 9194 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =  ( ( N  -  K )  +  1 ) )
1312fveq2d 5545 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ! `  ( ( N  -  K )  +  1 ) ) )
1412oveq2d 5890 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
156, 13, 143eqtr4d 2338 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
1615oveq1d 5889 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K
) ) )
17 faccl 11314 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
184, 17syl 15 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
1918nncnd 9778 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
20 nn0p1nn 10019 . . . . . . . . 9  |-  ( ( N  -  K )  e.  NN0  ->  ( ( N  -  K )  +  1 )  e.  NN )
214, 20syl 15 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  -  K
)  +  1 )  e.  NN )
2212, 21eqeltrd 2370 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
2322nncnd 9778 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
24 faccl 11314 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
2510, 24syl 15 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
2625nncnd 9778 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
2719, 23, 26mul32d 9038 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
2816, 27eqtrd 2328 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
293, 28oveq12d 5892 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  x.  ( N  + 
1 ) )  / 
( ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) )  x.  (
( N  +  1 )  -  K ) ) ) )
30 faccl 11314 . . . . . 6  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
311, 30syl 15 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  NN )
3231nncnd 9778 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  CC )
33 nn0p1nn 10019 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
341, 33syl 15 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  NN )
3534nncnd 9778 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
3618, 25nnmulcld 9809 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN )
37 nncn 9770 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC )
38 nnne0 9794 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =/=  0 )
3937, 38jca 518 . . . . 5  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
4036, 39syl 15 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
4122nnne0d 9806 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4223, 41jca 518 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  e.  CC  /\  ( ( N  + 
1 )  -  K
)  =/=  0 ) )
43 divmuldiv 9476 . . . 4  |-  ( ( ( ( ! `  N )  e.  CC  /\  ( N  +  1 )  e.  CC )  /\  ( ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC  /\  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =/=  0 )  /\  ( ( ( N  +  1 )  -  K )  e.  CC  /\  ( ( N  + 
1 )  -  K
)  =/=  0 ) ) )  ->  (
( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) )  x.  ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  =  ( ( ( ! `  N )  x.  ( N  + 
1 ) )  / 
( ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) )  x.  (
( N  +  1 )  -  K ) ) ) )
4432, 35, 40, 42, 43syl22anc 1183 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) )  x.  ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  =  ( ( ( ! `  N )  x.  ( N  + 
1 ) )  / 
( ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) )  x.  (
( N  +  1 )  -  K ) ) ) )
4529, 44eqtr4d 2331 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
46 fzelp1 10854 . . 3  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ( 0 ... ( N  +  1 ) ) )
47 bcval2 11334 . . 3  |-  ( K  e.  ( 0 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
4846, 47syl 15 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
49 bcval2 11334 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
5049oveq1d 5889 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  _C  K
)  x.  ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  =  ( ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
5145, 48, 503eqtr4d 2338 1  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ...cfz 10798   !cfa 11304    _C cbc 11331
This theorem is referenced by:  bcp1nk  11345  bcpasc  11349  bcp1ctr  20534  bcm1n  23048
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-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332
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