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Theorem bcp1n 12076
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 11469 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
2 facp1 12040 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 ( N  + 
1 ) )  =  ( ( ! `  N )  x.  ( N  +  1 ) ) )
31, 2syl 16 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
 N )  x.  ( N  +  1 ) ) )
4 fznn0sub 11474 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
5 facp1 12040 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( ( N  -  K )  +  1 ) )  =  ( ( ! `  ( N  -  K
) )  x.  (
( N  -  K
)  +  1 ) ) )
64, 5syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  -  K )  +  1 ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
71nn0cnd 10626 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  CC )
8 ax-1cn 9328 . . . . . . . . . 10  |-  1  e.  CC
98a1i 11 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  1  e.  CC )
10 elfznn0 11468 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
1110nn0cnd 10626 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
127, 9, 11addsubd 9728 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =  ( ( N  -  K )  +  1 ) )
1312fveq2d 5683 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ! `  ( ( N  -  K )  +  1 ) ) )
1412oveq2d 6096 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
156, 13, 143eqtr4d 2475 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
1615oveq1d 6095 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K
) ) )
17 faccl 12045 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
184, 17syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
1918nncnd 10326 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
20 nn0p1nn 10607 . . . . . . . . 9  |-  ( ( N  -  K )  e.  NN0  ->  ( ( N  -  K )  +  1 )  e.  NN )
214, 20syl 16 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  -  K
)  +  1 )  e.  NN )
2212, 21eqeltrd 2507 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
2322nncnd 10326 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
24 faccl 12045 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
2510, 24syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
2625nncnd 10326 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
2719, 23, 26mul32d 9567 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
2816, 27eqtrd 2465 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
293, 28oveq12d 6098 . . 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 12045 . . . . . 6  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
311, 30syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  NN )
3231nncnd 10326 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  CC )
33 nn0p1nn 10607 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
341, 33syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  NN )
3534nncnd 10326 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
3618, 25nnmulcld 10357 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN )
37 nncn 10318 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC )
38 nnne0 10342 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =/=  0 )
3937, 38jca 529 . . . . 5  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
4036, 39syl 16 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
4122nnne0d 10354 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4223, 41jca 529 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  e.  CC  /\  ( ( N  + 
1 )  -  K
)  =/=  0 ) )
43 divmuldiv 10019 . . . 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 1212 . . 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 2468 . 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 11492 . . 3  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ( 0 ... ( N  +  1 ) ) )
47 bcval2 12065 . . 3  |-  ( K  e.  ( 0 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
4846, 47syl 16 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
49 bcval2 12065 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
5049oveq1d 6095 . 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 2475 1  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   ` cfv 5406  (class class class)co 6080   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    - cmin 9583    / cdiv 9981   NNcn 10310   NN0cn0 10567   ...cfz 11424   !cfa 12035    _C cbc 12062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-seq 11791  df-fac 12036  df-bc 12063
This theorem is referenced by:  bcp1nk  12077  bcpasc  12081  bcp1ctr  22503  bcm1n  25902
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