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Theorem dvdsfac 12583
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 5525 . . . . 5  |-  ( x  =  K  ->  ( ! `  x )  =  ( ! `  K ) )
21breq2d 4035 . . . 4  |-  ( x  =  K  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  K )
) )
32imbi2d 307 . . 3  |-  ( x  =  K  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 K ) ) ) )
4 fveq2 5525 . . . . 5  |-  ( x  =  y  ->  ( ! `  x )  =  ( ! `  y ) )
54breq2d 4035 . . . 4  |-  ( x  =  y  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  y )
) )
65imbi2d 307 . . 3  |-  ( x  =  y  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 y ) ) ) )
7 fveq2 5525 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ! `  x )  =  ( ! `  ( y  +  1 ) ) )
87breq2d 4035 . . . 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 5525 . . . . 5  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1110breq2d 4035 . . . 4  |-  ( x  =  N  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  N )
) )
1211imbi2d 307 . . 3  |-  ( x  =  N  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 N ) ) ) )
13 nnm1nn0 10005 . . . . . . . 8  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
14 faccl 11298 . . . . . . . 8  |-  ( ( K  -  1 )  e.  NN0  ->  ( ! `
 ( K  - 
1 ) )  e.  NN )
1513, 14syl 15 . . . . . . 7  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  NN )
1615nnzd 10116 . . . . . 6  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  ZZ )
17 nnz 10045 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
18 dvdsmul2 12551 . . . . . 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 11297 . . . . 5  |-  ( K  e.  NN  ->  ( ! `  K )  =  ( ( ! `
 ( K  - 
1 ) )  x.  K ) )
2119, 20breqtrrd 4049 . . . 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 10264 . . . . . . . . . . . 12  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
25 uztrn 10244 . . . . . . . . . . . 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 10264 . . . . . . . . . . 11  |-  ( y  e.  NN  <->  y  e.  ( ZZ>= `  1 )
)
2826, 27sylibr 203 . . . . . . . . . 10  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN )
2928nnnn0d 10018 . . . . . . . . 9  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN0 )
30 faccl 11298 . . . . . . . . 9  |-  ( y  e.  NN0  ->  ( ! `
 y )  e.  NN )
3129, 30syl 15 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  NN )
3231nnzd 10116 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  ZZ )
3328nnzd 10116 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  ZZ )
3433peano2zd 10120 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  (
y  +  1 )  e.  ZZ )
35 dvdsmultr1 12563 . . . . . . 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 11293 . . . . . . . 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 4035 . . . . . 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 10276 . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   !cfa 11288    || cdivides 12531
This theorem is referenced by:  prmunb  12961  gexcl3  14898  wilth  20309  chtublem  20450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-fac 11289  df-dvds 12532
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