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Theorem dvds2lem 10352
Description: A lemma to assist theorems of  || with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds2lem.1  |-  ( ph  ->  ( I  e.  ZZ  /\  J  e.  ZZ ) )
dvds2lem.2  |-  ( ph  ->  ( K  e.  ZZ  /\  L  e.  ZZ ) )
dvds2lem.3  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
dvds2lem.4  |-  ( (
ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  Z  e.  ZZ )
dvds2lem.5  |-  ( (
ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L )  ->  ( Z  x.  M )  =  N ) )
Assertion
Ref Expression
dvds2lem  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  M  ||  N
) )
Distinct variable groups:    x, I, y   
x, J, y    x, K, y    x, L, y   
x, M, y    x, N, y    ph, x, y
Allowed substitution hints:    Z( x, y)

Proof of Theorem dvds2lem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dvds2lem.1 . . . . . 6  |-  ( ph  ->  ( I  e.  ZZ  /\  J  e.  ZZ ) )
2 dvds2lem.2 . . . . . 6  |-  ( ph  ->  ( K  e.  ZZ  /\  L  e.  ZZ ) )
3 divides 10342 . . . . . . 7  |-  ( ( I  e.  ZZ  /\  J  e.  ZZ )  ->  ( I  ||  J  <->  E. x  e.  ZZ  (
x  x.  I )  =  J ) )
4 divides 10342 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  ZZ )  ->  ( K  ||  L  <->  E. y  e.  ZZ  (
y  x.  K )  =  L ) )
53, 4bi2anan9 571 . . . . . 6  |-  ( ( ( I  e.  ZZ  /\  J  e.  ZZ )  /\  ( K  e.  ZZ  /\  L  e.  ZZ ) )  -> 
( ( I  ||  J  /\  K  ||  L
)  <->  ( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) ) )
61, 2, 5syl2anc 403 . . . . 5  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  <->  ( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) ) )
76biimpd 142 . . . 4  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  ( E. x  e.  ZZ  (
x  x.  I )  =  J  /\  E. y  e.  ZZ  (
y  x.  K )  =  L ) ) )
8 reeanv 2524 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  x.  I
)  =  J  /\  ( y  x.  K
)  =  L )  <-> 
( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) )
97, 8syl6ibr 160 . . 3  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L ) ) )
10 dvds2lem.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  Z  e.  ZZ )
11 dvds2lem.5 . . . . 5  |-  ( (
ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L )  ->  ( Z  x.  M )  =  N ) )
12 oveq1 5550 . . . . . . 7  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
1312eqeq1d 2090 . . . . . 6  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
1413rspcev 2702 . . . . 5  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
1510, 11, 14syl6an 1364 . . . 4  |-  ( (
ph  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L )  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
1615rexlimdvva 2485 . . 3  |-  ( ph  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  x.  I )  =  J  /\  ( y  x.  K )  =  L )  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
179, 16syld 44 . 2  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  E. z  e.  ZZ  ( z  x.  M )  =  N ) )
18 dvds2lem.3 . . 3  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
19 divides 10342 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2018, 19syl 14 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2117, 20sylibrd 167 1  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  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 3793  (class class class)co 5543    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-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
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 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-iota 4897  df-fv 4940  df-ov 5546  df-dvds 10341
This theorem is referenced by:  dvds2ln  10373  dvds2add  10374  dvds2sub  10375  dvdstr  10377
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