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Theorem dvdsfac 12904
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 5728 . . . . 5  |-  ( x  =  K  ->  ( ! `  x )  =  ( ! `  K ) )
21breq2d 4224 . . . 4  |-  ( x  =  K  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  K )
) )
32imbi2d 308 . . 3  |-  ( x  =  K  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 K ) ) ) )
4 fveq2 5728 . . . . 5  |-  ( x  =  y  ->  ( ! `  x )  =  ( ! `  y ) )
54breq2d 4224 . . . 4  |-  ( x  =  y  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  y )
) )
65imbi2d 308 . . 3  |-  ( x  =  y  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 y ) ) ) )
7 fveq2 5728 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( ! `  x )  =  ( ! `  ( y  +  1 ) ) )
87breq2d 4224 . . . 4  |-  ( x  =  ( y  +  1 )  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  ( y  +  1 ) ) ) )
98imbi2d 308 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 ( y  +  1 ) ) ) ) )
10 fveq2 5728 . . . . 5  |-  ( x  =  N  ->  ( ! `  x )  =  ( ! `  N ) )
1110breq2d 4224 . . . 4  |-  ( x  =  N  ->  ( K  ||  ( ! `  x )  <->  K  ||  ( ! `  N )
) )
1211imbi2d 308 . . 3  |-  ( x  =  N  ->  (
( K  e.  NN  ->  K  ||  ( ! `
 x ) )  <-> 
( K  e.  NN  ->  K  ||  ( ! `
 N ) ) ) )
13 nnm1nn0 10261 . . . . . . . 8  |-  ( K  e.  NN  ->  ( K  -  1 )  e.  NN0 )
14 faccl 11576 . . . . . . . 8  |-  ( ( K  -  1 )  e.  NN0  ->  ( ! `
 ( K  - 
1 ) )  e.  NN )
1513, 14syl 16 . . . . . . 7  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  NN )
1615nnzd 10374 . . . . . 6  |-  ( K  e.  NN  ->  ( ! `  ( K  -  1 ) )  e.  ZZ )
17 nnz 10303 . . . . . 6  |-  ( K  e.  NN  ->  K  e.  ZZ )
18 dvdsmul2 12872 . . . . . 6  |-  ( ( ( ! `  ( K  -  1 ) )  e.  ZZ  /\  K  e.  ZZ )  ->  K  ||  ( ( ! `  ( K  -  1 ) )  x.  K ) )
1916, 17, 18syl2anc 643 . . . . 5  |-  ( K  e.  NN  ->  K  ||  ( ( ! `  ( K  -  1
) )  x.  K
) )
20 facnn2 11575 . . . . 5  |-  ( K  e.  NN  ->  ( ! `  K )  =  ( ( ! `
 ( K  - 
1 ) )  x.  K ) )
2119, 20breqtrrd 4238 . . . 4  |-  ( K  e.  NN  ->  K  ||  ( ! `  K
) )
2221a1i 11 . . 3  |-  ( K  e.  ZZ  ->  ( K  e.  NN  ->  K 
||  ( ! `  K ) ) )
2317adantl 453 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  K  e.  ZZ )
24 elnnuz 10522 . . . . . . . . . . . 12  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
25 uztrn 10502 . . . . . . . . . . . 12  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  1 )
)  ->  y  e.  ( ZZ>= `  1 )
)
2624, 25sylan2b 462 . . . . . . . . . . 11  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  ( ZZ>= `  1 )
)
27 elnnuz 10522 . . . . . . . . . . 11  |-  ( y  e.  NN  <->  y  e.  ( ZZ>= `  1 )
)
2826, 27sylibr 204 . . . . . . . . . 10  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN )
2928nnnn0d 10274 . . . . . . . . 9  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  NN0 )
30 faccl 11576 . . . . . . . . 9  |-  ( y  e.  NN0  ->  ( ! `
 y )  e.  NN )
3129, 30syl 16 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  NN )
3231nnzd 10374 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  y )  e.  ZZ )
3328nnzd 10374 . . . . . . . 8  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  y  e.  ZZ )
3433peano2zd 10378 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  (
y  +  1 )  e.  ZZ )
35 dvdsmultr1 12884 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( ! `  y )  e.  ZZ  /\  (
y  +  1 )  e.  ZZ )  -> 
( K  ||  ( ! `  y )  ->  K  ||  ( ( ! `  y )  x.  ( y  +  1 ) ) ) )
3623, 32, 34, 35syl3anc 1184 . . . . . 6  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  y )  ->  K  ||  ( ( ! `  y )  x.  (
y  +  1 ) ) ) )
37 facp1 11571 . . . . . . . 8  |-  ( y  e.  NN0  ->  ( ! `
 ( y  +  1 ) )  =  ( ( ! `  y )  x.  (
y  +  1 ) ) )
3829, 37syl 16 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( ! `  ( y  +  1 ) )  =  ( ( ! `
 y )  x.  ( y  +  1 ) ) )
3938breq2d 4224 . . . . . 6  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  ( y  +  1 ) )  <->  K  ||  (
( ! `  y
)  x.  ( y  +  1 ) ) ) )
4036, 39sylibrd 226 . . . . 5  |-  ( ( y  e.  ( ZZ>= `  K )  /\  K  e.  NN )  ->  ( K  ||  ( ! `  y )  ->  K  ||  ( ! `  (
y  +  1 ) ) ) )
4140ex 424 . . . 4  |-  ( y  e.  ( ZZ>= `  K
)  ->  ( K  e.  NN  ->  ( K  ||  ( ! `  y
)  ->  K  ||  ( ! `  ( y  +  1 ) ) ) ) )
4241a2d 24 . . 3  |-  ( y  e.  ( ZZ>= `  K
)  ->  ( ( K  e.  NN  ->  K 
||  ( ! `  y ) )  -> 
( K  e.  NN  ->  K  ||  ( ! `
 ( y  +  1 ) ) ) ) )
433, 6, 9, 12, 22, 42uzind4 10534 . 2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( K  e.  NN  ->  K  ||  ( ! `  N )
) )
4443impcom 420 1  |-  ( ( K  e.  NN  /\  N  e.  ( ZZ>= `  K ) )  ->  K  ||  ( ! `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   !cfa 11566    || cdivides 12852
This theorem is referenced by:  prmunb  13282  gexcl3  15221  wilth  20854  chtublem  20995
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-fac 11567  df-dvds 12853
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