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Theorem dvds1lem 12540
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 5865 . . . . . 6  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
43eqeq1d 2291 . . . . 5  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
54rspcev 2884 . . . 4  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
61, 2, 5ee12an 1353 . . 3  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
76rexlimdva 2667 . 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 12533 . . 3  |-  ( ( J  e.  ZZ  /\  K  e.  ZZ )  ->  ( J  ||  K  <->  E. x  e.  ZZ  (
x  x.  J )  =  K ) )
108, 9syl 15 . 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 12533 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
1311, 12syl 15 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
147, 10, 133imtr4d 259 1  |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023  (class class class)co 5858    x. cmul 8742   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  negdvdsb  12545  dvdsnegb  12546  muldvds1  12553  muldvds2  12554  dvdscmul  12555  dvdsmulc  12556  dvdscmulr  12557  dvdsmulcr  12558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-dvds 12532
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