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Theorem dvds2lem 12557
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 12549 . . . . . . 7  |-  ( ( I  e.  ZZ  /\  J  e.  ZZ )  ->  ( I  ||  J  <->  E. x  e.  ZZ  (
x  x.  I )  =  J ) )
4 divides 12549 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  ZZ )  ->  ( K  ||  L  <->  E. y  e.  ZZ  (
y  x.  K )  =  L ) )
53, 4bi2anan9 843 . . . . . 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 642 . . . . 5  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  <->  ( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) ) )
76biimpd 198 . . . 4  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  ( E. x  e.  ZZ  (
x  x.  I )  =  J  /\  E. y  e.  ZZ  (
y  x.  K )  =  L ) ) )
8 reeanv 2720 . . . 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 218 . . 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 5881 . . . . . . 7  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
1312eqeq1d 2304 . . . . . 6  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
1413rspcev 2897 . . . . 5  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
1510, 11, 14ee12an 1353 . . . 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 2687 . . 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 40 . 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 12549 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2018, 19syl 15 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2117, 20sylibrd 225 1  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  M  ||  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874    x. cmul 8758   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  dvds2ln  12575  dvds2add  12576  dvds2sub  12577  dvdstr  12578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279  df-ov 5877  df-dvds 12548
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