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Theorem bcp1n 12375
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 11782 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  N  e.  NN0 )
2 facp1 12339 . . . . 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 11726 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  -  K )  e.  NN0 )
5 facp1 12339 . . . . . . . 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 10861 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  CC )
8 1cnd 9615 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  1  e.  CC )
9 elfznn0 11781 . . . . . . . . . 10  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
109nn0cnd 10861 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
117, 8, 10addsubd 9957 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =  ( ( N  -  K )  +  1 ) )
1211fveq2d 5860 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ! `  ( ( N  -  K )  +  1 ) ) )
1311oveq2d 6297 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  -  K )  +  1 ) ) )
146, 12, 133eqtr4d 2494 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( ( N  +  1 )  -  K ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
1514oveq1d 6296 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K
) ) )
16 faccl 12344 . . . . . . . 8  |-  ( ( N  -  K )  e.  NN0  ->  ( ! `
 ( N  -  K ) )  e.  NN )
174, 16syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  NN )
1817nncnd 10559 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  ( N  -  K ) )  e.  CC )
19 nn0p1nn 10842 . . . . . . . . 9  |-  ( ( N  -  K )  e.  NN0  ->  ( ( N  -  K )  +  1 )  e.  NN )
204, 19syl 16 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  (
( N  -  K
)  +  1 )  e.  NN )
2111, 20eqeltrd 2531 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
2221nncnd 10559 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
23 faccl 12344 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( ! `
 K )  e.  NN )
249, 23syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  NN )
2524nncnd 10559 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  K )  e.  CC )
2618, 22, 25mul32d 9793 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
2715, 26eqtrd 2484 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  (
( N  +  1 )  -  K ) )  x.  ( ! `
 K ) )  =  ( ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) )  x.  ( ( N  + 
1 )  -  K
) ) )
283, 27oveq12d 6299 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  x.  ( N  + 
1 ) )  / 
( ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) )  x.  (
( N  +  1 )  -  K ) ) ) )
29 faccl 12344 . . . . . 6  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
301, 29syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  NN )
3130nncnd 10559 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( ! `  N )  e.  CC )
32 nn0p1nn 10842 . . . . . 6  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
331, 32syl 16 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  NN )
3433nncnd 10559 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
3517, 24nnmulcld 10590 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN )
36 nncn 10551 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  CC )
37 nnne0 10575 . . . . . 6  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  =/=  0 )
3836, 37jca 532 . . . . 5  |-  ( ( ( ! `  ( N  -  K )
)  x.  ( ! `
 K ) )  e.  NN  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
3935, 38syl 16 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  e.  CC  /\  ( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
)  =/=  0 ) )
4021nnne0d 10587 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4122, 40jca 532 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  e.  CC  /\  ( ( N  + 
1 )  -  K
)  =/=  0 ) )
42 divmuldiv 10251 . . . 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 ) ) ) )
4331, 34, 39, 41, 42syl22anc 1230 . . 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 ) ) ) )
4428, 43eqtr4d 2487 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( ! `  ( N  +  1 ) )  /  ( ( ! `  ( ( N  +  1 )  -  K ) )  x.  ( ! `  K ) ) )  =  ( ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
45 fzelp1 11742 . . 3  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ( 0 ... ( N  +  1 ) ) )
46 bcval2 12364 . . 3  |-  ( K  e.  ( 0 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
4745, 46syl 16 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( ! `
 ( N  + 
1 ) )  / 
( ( ! `  ( ( N  + 
1 )  -  K
) )  x.  ( ! `  K )
) ) )
48 bcval2 12364 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) )
4948oveq1d 6296 . 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 ) ) ) )
5044, 47, 493eqtr4d 2494 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 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810    / cdiv 10213   NNcn 10543   NN0cn0 10802   ...cfz 11682   !cfa 12334    _C cbc 12361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11092  df-fz 11683  df-seq 12089  df-fac 12335  df-bc 12362
This theorem is referenced by:  bcp1nk  12376  bcpasc  12380  bcp1ctr  23530  bcm1n  27576
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