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Theorem bcp1n 12358
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 11767 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
2 facp1 12322 . . . . 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 11712 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
5 facp1 12322 . . . . . . . 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 10850 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  CC )
8 ax-1cn 9546 . . . . . . . . . 10  |-  1  e.  CC
98a1i 11 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  1  e.  CC )
10 elfznn0 11766 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
1110nn0cnd 10850 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
127, 9, 11addsubd 9947 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =  ( ( N  -  K )  +  1 ) )
1312fveq2d 5868 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ! `  ( ( N  -  K )  +  1 ) ) )
1412oveq2d 6298 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
156, 13, 143eqtr4d 2518 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
1615oveq1d 6297 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K
) ) )
17 faccl 12327 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
184, 17syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
1918nncnd 10548 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
20 nn0p1nn 10831 . . . . . . . . 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 2555 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
2322nncnd 10548 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
24 faccl 12327 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
2510, 24syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
2625nncnd 10548 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
2719, 23, 26mul32d 9785 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
2816, 27eqtrd 2508 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
293, 28oveq12d 6300 . . 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 12327 . . . . . 6  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
311, 30syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  NN )
3231nncnd 10548 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  CC )
33 nn0p1nn 10831 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
341, 33syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  NN )
3534nncnd 10548 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
3618, 25nnmulcld 10579 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN )
37 nncn 10540 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC )
38 nnne0 10564 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =/=  0 )
3937, 38jca 532 . . . . 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 10576 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4223, 41jca 532 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  e.  CC  /\  ( ( N  + 
1 )  -  K
)  =/=  0 ) )
43 divmuldiv 10240 . . . 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 1229 . . 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 2511 . 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 11728 . . 3  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ( 0 ... ( N  +  1 ) ) )
47 bcval2 12347 . . 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 12347 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
5049oveq1d 6297 . 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 2518 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 1379    e. wcel 1767    =/= wne 2662   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801    / cdiv 10202   NNcn 10532   NN0cn0 10791   ...cfz 11668   !cfa 12317    _C cbc 12344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-fac 12318  df-bc 12345
This theorem is referenced by:  bcp1nk  12359  bcpasc  12363  bcp1ctr  23282  bcm1n  27268
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