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Theorem dvds2lem 11432
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 11422 . . . . . . 7  |-  ( ( I  e.  ZZ  /\  J  e.  ZZ )  ->  ( I  ||  J  <->  E. x  e.  ZZ  (
x  x.  I )  =  J ) )
4 divides 11422 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  ZZ )  ->  ( K  ||  L  <->  E. y  e.  ZZ  (
y  x.  K )  =  L ) )
53, 4bi2anan9 580 . . . . . 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 408 . . . . 5  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  <->  ( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) ) )
76biimpd 143 . . . 4  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  ( E. x  e.  ZZ  (
x  x.  I )  =  J  /\  E. y  e.  ZZ  (
y  x.  K )  =  L ) ) )
8 reeanv 2577 . . . 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 161 . . 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 5749 . . . . . . 7  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
1312eqeq1d 2126 . . . . . 6  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
1413rspcev 2763 . . . . 5  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
1510, 11, 14syl6an 1395 . . . 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 2534 . . 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 45 . 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 11422 . . 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 168 1  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  M  ||  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742    x. cmul 7593   ZZcz 9022    || cdvds 11420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-iota 5058  df-fv 5101  df-ov 5745  df-dvds 11421
This theorem is referenced by:  dvds2ln  11453  dvds2add  11454  dvds2sub  11455  dvdstr  11457
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