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Theorem dvds1lem 10340
Description: A lemma to assist theorems of  || with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds1lem.1  |-  ( ph  ->  ( J  e.  ZZ  /\  K  e.  ZZ ) )
dvds1lem.2  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
dvds1lem.3  |-  ( (
ph  /\  x  e.  ZZ )  ->  Z  e.  ZZ )
dvds1lem.4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  ( Z  x.  M )  =  N ) )
Assertion
Ref Expression
dvds1lem  |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
Distinct variable groups:    x, J    x, K    x, M    x, N    ph, x
Allowed substitution hint:    Z( x)

Proof of Theorem dvds1lem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dvds1lem.3 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  Z  e.  ZZ )
2 dvds1lem.4 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  ( Z  x.  M )  =  N ) )
3 oveq1 5544 . . . . . 6  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
43eqeq1d 2090 . . . . 5  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
54rspcev 2702 . . . 4  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
61, 2, 5syl6an 1364 . . 3  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
76rexlimdva 2478 . 2  |-  ( ph  ->  ( E. x  e.  ZZ  ( x  x.  J )  =  K  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N ) )
8 dvds1lem.1 . . 3  |-  ( ph  ->  ( J  e.  ZZ  /\  K  e.  ZZ ) )
9 divides 10331 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  ||  K  <->  E. x  e.  ZZ  (
x  x.  J )  =  K ) )
108, 9syl 14 . 2  |-  ( ph  ->  ( J  ||  K  <->  E. x  e.  ZZ  (
x  x.  J )  =  K ) )
11 dvds1lem.2 . . 3  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
12 divides 10331 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
1311, 12syl 14 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
147, 10, 133imtr4d 201 1  |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3787  (class class class)co 5537    x. cmul 7037   ZZcz 8421    || cdvds 10329
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-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 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  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-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-iota 4891  df-fv 4934  df-ov 5540  df-dvds 10330
This theorem is referenced by:  negdvdsb  10345  dvdsnegb  10346  muldvds1  10354  muldvds2  10355  dvdscmul  10356  dvdsmulc  10357  dvdscmulr  10358  dvdsmulcr  10359
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