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Theorem muldvds1 14092
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
muldvds1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  K 
||  N ) )

Proof of Theorem muldvds1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 zmulcl 10908 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
21anim1i 566 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  e.  ZZ  /\  N  e.  ZZ ) )
323impa 1189 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )
)
4 3simpb 992 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
5 zmulcl 10908 . . . 4  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  x.  M
)  e.  ZZ )
65ancoms 451 . . 3  |-  ( ( M  e.  ZZ  /\  x  e.  ZZ )  ->  ( x  x.  M
)  e.  ZZ )
763ad2antl2 1157 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  M )  e.  ZZ )
8 zcn 10865 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 10865 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
10 zcn 10865 . . . . . . . 8  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 mulass 9569 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( x  x.  ( K  x.  M
) ) )
12 mul32 9736 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
( x  x.  K
)  x.  M )  =  ( ( x  x.  M )  x.  K ) )
1311, 12eqtr3d 2497 . . . . . . . 8  |-  ( ( x  e.  CC  /\  K  e.  CC  /\  M  e.  CC )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
148, 9, 10, 13syl3an 1268 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  K  e.  ZZ  /\  M  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
15143coml 1201 . . . . . 6  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
x  x.  ( K  x.  M ) )  =  ( ( x  x.  M )  x.  K ) )
16153expa 1194 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( ( x  x.  M
)  x.  K ) )
17163adantl3 1152 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( x  x.  ( K  x.  M
) )  =  ( ( x  x.  M
)  x.  K ) )
1817eqeq1d 2456 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  ( K  x.  M ) )  =  N  <->  ( ( x  x.  M )  x.  K )  =  N ) )
1918biimpd 207 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  ( K  x.  M ) )  =  N  ->  ( (
x  x.  M )  x.  K )  =  N ) )
203, 4, 7, 19dvds1lem 14079 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  x.  M
)  ||  N  ->  K 
||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439  (class class class)co 6270   CCcc 9479    x. cmul 9486   ZZcz 10860    || cdvds 14070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-dvds 14071
This theorem is referenced by:  3dvds  14134  odmulg  16777  jm2.20nn  31178  jm2.27c  31188  nzss  31463  etransclem28  32284
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