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Theorem dvds1lem 11352
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 5735 . . . . . 6  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
43eqeq1d 2123 . . . . 5  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
54rspcev 2760 . . . 4  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
61, 2, 5syl6an 1393 . . 3  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( x  x.  J )  =  K  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
76rexlimdva 2523 . 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 11343 . . 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 11343 . . 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 202 1  |-  ( ph  ->  ( J  ||  K  ->  M  ||  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728    x. cmul 7552   ZZcz 8958    || cdvds 11341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-iota 5046  df-fv 5089  df-ov 5731  df-dvds 11342
This theorem is referenced by:  negdvdsb  11357  dvdsnegb  11358  muldvds1  11366  muldvds2  11367  dvdscmul  11368  dvdsmulc  11369  dvdscmulr  11370  dvdsmulcr  11371
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