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Theorem dvds1lem 13873
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 6302 . . . . . 6  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
43eqeq1d 2469 . . . . 5  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
54rspcev 3219 . . . 4  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
61, 2, 5syl6an 545 . . 3  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
76rexlimdva 2959 . 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 13866 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  ||  K  <->  E. x  e.  ZZ  (
x  x.  J )  =  K ) )
108, 9syl 16 . 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 13866 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
1311, 12syl 16 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
147, 10, 133imtr4d 268 1  |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453  (class class class)co 6295    x. cmul 9509   ZZcz 10876    || cdivides 13864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-iota 5557  df-fv 5602  df-ov 6298  df-dvds 13865
This theorem is referenced by:  negdvdsb  13878  dvdsnegb  13879  muldvds1  13886  muldvds2  13887  dvdscmul  13888  dvdsmulc  13889  dvdscmulr  13890  dvdsmulcr  13891
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