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Theorem dvdsmulc 12653
Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdsmulc  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )

Proof of Theorem dvdsmulc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 3simpc 954 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 zmulcl 10158 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K
)  e.  ZZ )
323adant2 974 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  x.  K )  e.  ZZ )
4 zmulcl 10158 . . . . . 6  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K
)  e.  ZZ )
543adant1 973 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  x.  K )  e.  ZZ )
63, 5jca 518 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M  x.  K
)  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
763comr 1159 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  x.  K
)  e.  ZZ  /\  ( N  x.  K
)  e.  ZZ ) )
8 simpr 447 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
9 zcn 10121 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
10 zcn 10121 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  M  e.  CC )
11 zcn 10121 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
12 mulass 8915 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  M  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
139, 10, 11, 12syl3an 1224 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  K  e.  ZZ )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
14133com13 1156 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  x  e.  ZZ )  ->  (
( x  x.  M
)  x.  K )  =  ( x  x.  ( M  x.  K
) ) )
15143expa 1151 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K ) ) )
16153adantl3 1113 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  x.  K )  =  ( x  x.  ( M  x.  K ) ) )
17 oveq1 5952 . . . . 5  |-  ( ( x  x.  M )  =  N  ->  (
( x  x.  M
)  x.  K )  =  ( N  x.  K ) )
1816, 17sylan9req 2411 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  /\  (
x  x.  M )  =  N )  -> 
( x  x.  ( M  x.  K )
)  =  ( N  x.  K ) )
1918ex 423 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x  x.  M )  =  N  ->  ( x  x.  ( M  x.  K
) )  =  ( N  x.  K ) ) )
201, 7, 8, 19dvds1lem 12637 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
21203coml 1158 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  N  ->  ( M  x.  K )  ||  ( N  x.  K
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   CCcc 8825    x. cmul 8832   ZZcz 10116    || cdivides 12628
This theorem is referenced by:  dvdsmulcr  12655  coprmdvds2  12879  mulgcddvds  12880  rpmulgcd2  12881  pcpremul  12993  odadd2  15240  ablfacrp2  15401  znrrg  16625  dvdsmulf1o  20546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-ltxr 8962  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-dvds 12629
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