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Theorem dvdsfac 12599
Description: A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
Assertion
Ref Expression
dvdsfac  |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>= `  K ) )  ->  K  ||  ( ! `  N ) )

Proof of Theorem dvdsfac
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . 5  |-  ( x  =  K  ->  ( ! `  x )  =  ( ! `  K ) )
21breq2d 4051 . . . 4  |-  ( x  =  K  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  K )
) )
32imbi2d 307 . . 3  |-  ( x  =  K  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 K ) ) ) )
4 fveq2 5541 . . . . 5  |-  ( x  =  y  ->  ( ! `  x )  =  ( ! `  y ) )
54breq2d 4051 . . . 4  |-  ( x  =  y  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  y )
) )
65imbi2d 307 . . 3  |-  ( x  =  y  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 y ) ) ) )
7 fveq2 5541 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ! `  x )  =  ( ! `  ( y  +  1 ) ) )
87breq2d 4051 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  ( y  +  1 ) ) ) )
98imbi2d 307 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 ( y  +  1 ) ) ) ) )
10 fveq2 5541 . . . . 5  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1110breq2d 4051 . . . 4  |-  ( x  =  N  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  N )
) )
1211imbi2d 307 . . 3  |-  ( x  =  N  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 N ) ) ) )
13 nnm1nn0 10021 . . . . . . . 8  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
14 faccl 11314 . . . . . . . 8  |-  ( ( K  -  1 )  e.  NN0  ->  ( ! `
 ( K  - 
1 ) )  e.  NN )
1513, 14syl 15 . . . . . . 7  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  NN )
1615nnzd 10132 . . . . . 6  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  ZZ )
17 nnz 10061 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
18 dvdsmul2 12567 . . . . . 6  |-  ( ( ( ! `  ( K  -  1 ) )  e.  ZZ  /\  K  e.  ZZ )  ->  K  ||  ( ( ! `  ( K  -  1 ) )  x.  K ) )
1916, 17, 18syl2anc 642 . . . . 5  |-  ( K  e.  NN  ->  K  ||  ( ( ! `  ( K  -  1
) )  x.  K
) )
20 facnn2 11313 . . . . 5  |-  ( K  e.  NN  ->  ( ! `  K )  =  ( ( ! `
 ( K  - 
1 ) )  x.  K ) )
2119, 20breqtrrd 4065 . . . 4  |-  ( K  e.  NN  ->  K  ||  ( ! `  K
) )
2221a1i 10 . . 3  |-  ( K  e.  ZZ  ->  ( K  e.  NN  ->  K 
||  ( ! `  K ) ) )
2317adantl 452 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  K  e.  ZZ )
24 elnnuz 10280 . . . . . . . . . . . 12  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
25 uztrn 10260 . . . . . . . . . . . 12  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  1 )
)  ->  y  e.  ( ZZ>= `  1 )
)
2624, 25sylan2b 461 . . . . . . . . . . 11  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  ( ZZ>= `  1 )
)
27 elnnuz 10280 . . . . . . . . . . 11  |-  ( y  e.  NN  <->  y  e.  ( ZZ>= `  1 )
)
2826, 27sylibr 203 . . . . . . . . . 10  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN )
2928nnnn0d 10034 . . . . . . . . 9  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN0 )
30 faccl 11314 . . . . . . . . 9  |-  ( y  e.  NN0  ->  ( ! `
 y )  e.  NN )
3129, 30syl 15 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  NN )
3231nnzd 10132 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  ZZ )
3328nnzd 10132 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  ZZ )
3433peano2zd 10136 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  (
y  +  1 )  e.  ZZ )
35 dvdsmultr1 12579 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( ! `  y )  e.  ZZ  /\  (
y  +  1 )  e.  ZZ )  -> 
( K  ||  ( ! `  y )  ->  K  ||  ( ( ! `  y )  x.  ( y  +  1 ) ) ) )
3623, 32, 34, 35syl3anc 1182 . . . . . 6  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  y )  ->  K  ||  ( ( ! `  y )  x.  (
y  +  1 ) ) ) )
37 facp1 11309 . . . . . . . 8  |-  ( y  e.  NN0  ->  ( ! `
 ( y  +  1 ) )  =  ( ( ! `  y )  x.  (
y  +  1 ) ) )
3829, 37syl 15 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  ( y  +  1 ) )  =  ( ( ! `
 y )  x.  ( y  +  1 ) ) )
3938breq2d 4051 . . . . . 6  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  ( y  +  1 ) )  <->  K  ||  (
( ! `  y
)  x.  ( y  +  1 ) ) ) )
4036, 39sylibrd 225 . . . . 5  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  y )  ->  K  ||  ( ! `  (
y  +  1 ) ) ) )
4140ex 423 . . . 4  |-  ( y  e.  ( ZZ>= `  K
)  ->  ( K  e.  NN  ->  ( K  ||  ( ! `  y
)  ->  K  ||  ( ! `  ( y  +  1 ) ) ) ) )
4241a2d 23 . . 3  |-  ( y  e.  ( ZZ>= `  K
)  ->  ( ( K  e.  NN  ->  K 
||  ( ! `  y ) )  -> 
( K  e.  NN  ->  K  ||  ( ! `
 ( y  +  1 ) ) ) ) )
433, 6, 9, 12, 22, 42uzind4 10292 . 2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( K  e.  NN  ->  K  ||  ( ! `  N )
) )
4443impcom 419 1  |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>= `  K ) )  ->  K  ||  ( ! `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   !cfa 11304    || cdivides 12547
This theorem is referenced by:  prmunb  12977  gexcl3  14914  wilth  20325  chtublem  20466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-fac 11305  df-dvds 12548
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