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Theorem bcp1n 10695
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 10071 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
2 facp1 10664 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 ( N  + 
1 ) )  =  ( ( ! `  N )  x.  ( N  +  1 ) ) )
31, 2syl 14 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
 N )  x.  ( N  +  1 ) ) )
4 fznn0sub 10013 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
5 facp1 10664 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( ( N  -  K )  +  1 ) )  =  ( ( ! `  ( N  -  K
) )  x.  (
( N  -  K
)  +  1 ) ) )
64, 5syl 14 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  -  K )  +  1 ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
71nn0cnd 9190 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  CC )
8 1cnd 7936 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  1  e.  CC )
9 elfznn0 10070 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
109nn0cnd 9190 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
117, 8, 10addsubd 8251 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =  ( ( N  -  K )  +  1 ) )
1211fveq2d 5500 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ! `  ( ( N  -  K )  +  1 ) ) )
1311oveq2d 5869 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
146, 12, 133eqtr4d 2213 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
1514oveq1d 5868 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K
) ) )
164faccld 10670 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
1716nncnd 8892 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
18 nn0p1nn 9174 . . . . . . . . 9  |-  ( ( N  -  K )  e.  NN0  ->  ( ( N  -  K )  +  1 )  e.  NN )
194, 18syl 14 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  -  K
)  +  1 )  e.  NN )
2011, 19eqeltrd 2247 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
2120nncnd 8892 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
229faccld 10670 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
2322nncnd 8892 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
2417, 21, 23mul32d 8072 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
2515, 24eqtrd 2203 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
263, 25oveq12d 5871 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  x.  ( N  + 
1 ) )  / 
( ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) )  x.  (
( N  +  1 )  -  K ) ) ) )
271faccld 10670 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  NN )
2827nncnd 8892 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  CC )
2916, 22nnmulcld 8927 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN )
3029nncnd 8892 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC )
31 nn0p1nn 9174 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
321, 31syl 14 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  NN )
3332nncnd 8892 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
3429nnap0d 8924 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) #  0 )
3520nnap0d 8924 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K ) #  0 )
3628, 30, 33, 21, 34, 35divmuldivapd 8749 . . 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 ) ) ) )
3726, 36eqtr4d 2206 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
38 fzelp1 10030 . . 3  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ( 0 ... ( N  +  1 ) ) )
39 bcval2 10684 . . 3  |-  ( K  e.  ( 0 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
4038, 39syl 14 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
41 bcval2 10684 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
4241oveq1d 5868 . 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 ) ) ) )
4337, 40, 423eqtr4d 2213 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    = wceq 1348    e. wcel 2141   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    - cmin 8090    / cdiv 8589   NNcn 8878   NN0cn0 9135   ...cfz 9965   !cfa 10659    _C cbc 10681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-fz 9966  df-seqfrec 10402  df-fac 10660  df-bc 10682
This theorem is referenced by:  bcp1nk  10696  bcpasc  10700
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