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Theorem dvdstr 10377
Description: The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
dvdstr  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  M  ||  N )  ->  K  ||  N
) )

Proof of Theorem dvdstr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpa 936 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  M  e.  ZZ ) )
2 3simpc 938 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
3 3simpb 937 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ZZ  /\  N  e.  ZZ ) )
4 zmulcl 8485 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
54adantl 271 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  x.  y )  e.  ZZ )
6 oveq2 5551 . . . . 5  |-  ( ( x  x.  K )  =  M  ->  (
y  x.  ( x  x.  K ) )  =  ( y  x.  M ) )
76adantr 270 . . . 4  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  M
)  =  N )  ->  ( y  x.  ( x  x.  K
) )  =  ( y  x.  M ) )
8 eqeq2 2091 . . . . 5  |-  ( ( y  x.  M )  =  N  ->  (
( y  x.  (
x  x.  K ) )  =  ( y  x.  M )  <->  ( y  x.  ( x  x.  K
) )  =  N ) )
98adantl 271 . . . 4  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  M
)  =  N )  ->  ( ( y  x.  ( x  x.  K ) )  =  ( y  x.  M
)  <->  ( y  x.  ( x  x.  K
) )  =  N ) )
107, 9mpbid 145 . . 3  |-  ( ( ( x  x.  K
)  =  M  /\  ( y  x.  M
)  =  N )  ->  ( y  x.  ( x  x.  K
) )  =  N )
11 zcn 8437 . . . . . . . 8  |-  ( x  e.  ZZ  ->  x  e.  CC )
12 zcn 8437 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
13 zcn 8437 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
14 mulass 7166 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  y
)  x.  K )  =  ( x  x.  ( y  x.  K
) ) )
15 mul12 7304 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
x  x.  ( y  x.  K ) )  =  ( y  x.  ( x  x.  K
) ) )
1614, 15eqtrd 2114 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  K  e.  CC )  ->  (
( x  x.  y
)  x.  K )  =  ( y  x.  ( x  x.  K
) ) )
1711, 12, 13, 16syl3an 1212 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ  /\  K  e.  ZZ )  ->  (
( x  x.  y
)  x.  K )  =  ( y  x.  ( x  x.  K
) ) )
18173comr 1147 . . . . . 6  |-  ( ( K  e.  ZZ  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( x  x.  y
)  x.  K )  =  ( y  x.  ( x  x.  K
) ) )
19183expb 1140 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  x.  y )  x.  K )  =  ( y  x.  (
x  x.  K ) ) )
20193ad2antl1 1101 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  x.  y )  x.  K )  =  ( y  x.  (
x  x.  K ) ) )
2120eqeq1d 2090 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  y
)  x.  K )  =  N  <->  ( y  x.  ( x  x.  K
) )  =  N ) )
2210, 21syl5ibr 154 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
( x  x.  K
)  =  M  /\  ( y  x.  M
)  =  N )  ->  ( ( x  x.  y )  x.  K )  =  N ) )
231, 2, 3, 5, 22dvds2lem 10352 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  M  ||  N )  ->  K  ||  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041    x. cmul 7048   ZZcz 8432    || cdvds 10340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-dvds 10341
This theorem is referenced by:  dvdsmultr1  10378  dvdsmultr2  10380  4dvdseven  10461  dvdsgcdb  10546  dvdsmulgcd  10558  gcddvdslcm  10599  lcmgcdeq  10609  lcmdvdsb  10610  mulgcddvds  10620  rpmulgcd2  10621  rpdvds  10625  exprmfct  10663  rpexp  10676
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