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Theorem bcp1nk 11600
Description: The proportion of one binomial coefficient to another with  N and  K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
Assertion
Ref Expression
bcp1nk  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )

Proof of Theorem bcp1nk
StepHypRef Expression
1 elfzel1 11050 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  0  e.  ZZ )
2 elfzel2 11049 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
3 elfzelz 11051 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
4 1z 10303 . . . . . . 7  |-  1  e.  ZZ
54a1i 11 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  1  e.  ZZ )
6 fzaddel 11079 . . . . . 6  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( K  e.  ( 0 ... N )  <-> 
( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  + 
1 ) ) ) )
71, 2, 3, 5, 6syl22anc 1185 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( K  e.  ( 0 ... N )  <->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) ) )
87ibi 233 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( ( 0  +  1 ) ... ( N  +  1 ) ) )
9 1e0p1 10402 . . . . 5  |-  1  =  ( 0  +  1 )
109oveq1i 6083 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  =  ( ( 0  +  1 ) ... ( N  +  1 ) )
118, 10syl6eleqr 2526 . . 3  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
12 bcm1k 11598 . . 3  |-  ( ( K  +  1 )  e.  ( 1 ... ( N  +  1 ) )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( ( N  +  1 )  _C  ( ( K  +  1 )  - 
1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  + 
1 )  -  1 ) )  /  ( K  +  1 ) ) ) )
1311, 12syl 16 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( ( N  +  1 )  _C  ( ( K  +  1 )  - 
1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  + 
1 )  -  1 ) )  /  ( K  +  1 ) ) ) )
143zcnd 10368 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
15 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
16 pncan 9303 . . . . . . 7  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  + 
1 )  -  1 )  =  K )
1714, 15, 16sylancl 644 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( K  +  1 )  -  1 )  =  K )
1817oveq2d 6089 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  _C  K ) )
19 bcp1n 11599 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2018, 19eqtrd 2467 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( ( K  +  1 )  -  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) ) ) )
2117oveq2d 6089 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  =  ( ( N  +  1 )  -  K ) )
2221oveq1d 6088 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  (
( K  +  1 )  -  1 ) )  /  ( K  +  1 ) )  =  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) )
2320, 22oveq12d 6091 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  _C  (
( K  +  1 )  -  1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  /  ( K  + 
1 ) ) )  =  ( ( ( N  _C  K )  x.  ( ( N  +  1 )  / 
( ( N  + 
1 )  -  K
) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  +  1 ) ) ) )
24 bcrpcl 11591 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
2524rpcnd 10642 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  e.  CC )
262peano2zd 10370 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  ZZ )
2726zred 10367 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  RR )
283zred 10367 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
292zred 10367 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
30 elfzle2 11053 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
3129ltp1d 9933 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  N  <  ( N  +  1 ) )
3228, 29, 27, 30, 31lelttrd 9220 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  <  ( N  +  1 ) )
33 znnsub 10314 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  ( N  +  1
)  e.  ZZ )  ->  ( K  < 
( N  +  1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
343, 26, 33syl2anc 643 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  ( K  <  ( N  + 
1 )  <->  ( ( N  +  1 )  -  K )  e.  NN ) )
3532, 34mpbid 202 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  NN )
3627, 35nndivred 10040 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  RR )
3736recnd 9106 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  e.  CC )
3835nnred 10007 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  RR )
39 elfznn0 11075 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  NN0 )
40 nn0p1nn 10251 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( K  +  1 )  e.  NN )
4139, 40syl 16 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  NN )
4238, 41nndivred 10040 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  RR )
4342recnd 9106 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) )  e.  CC )
4425, 37, 43mulassd 9103 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  _C  K )  x.  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( ( N  +  1 )  / 
( ( N  + 
1 )  -  K
) )  x.  (
( ( N  + 
1 )  -  K
)  /  ( K  +  1 ) ) ) ) )
4526zcnd 10368 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  +  1 )  e.  CC )
4635nncnd 10008 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  e.  CC )
4741nncnd 10008 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  e.  CC )
4835nnne0d 10036 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  -  K )  =/=  0 )
4941nnne0d 10036 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( K  +  1 )  =/=  0 )
5045, 46, 47, 48, 49dmdcan2d 9812 . . . . 5  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  /  (
( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  +  1 )  / 
( K  +  1 ) ) )
5150oveq2d 6089 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  (
( N  _C  K
)  x.  ( ( ( N  +  1 )  /  ( ( N  +  1 )  -  K ) )  x.  ( ( ( N  +  1 )  -  K )  / 
( K  +  1 ) ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5244, 51eqtrd 2467 . . 3  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  _C  K )  x.  (
( N  +  1 )  /  ( ( N  +  1 )  -  K ) ) )  x.  ( ( ( N  +  1 )  -  K )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5323, 52eqtrd 2467 . 2  |-  ( K  e.  ( 0 ... N )  ->  (
( ( N  + 
1 )  _C  (
( K  +  1 )  -  1 ) )  x.  ( ( ( N  +  1 )  -  ( ( K  +  1 )  -  1 ) )  /  ( K  + 
1 ) ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
5413, 53eqtrd 2467 1  |-  ( K  e.  ( 0 ... N )  ->  (
( N  +  1 )  _C  ( K  +  1 ) )  =  ( ( N  _C  K )  x.  ( ( N  + 
1 )  /  ( K  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ...cfz 11035    _C cbc 11585
This theorem is referenced by:  sylow1lem1  15224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-fac 11559  df-bc 11586
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