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Theorem dvds2lem 13566
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 13558 . . . . . . 7  |-  ( ( I  e.  ZZ  /\  J  e.  ZZ )  ->  ( I  ||  J  <->  E. x  e.  ZZ  (
x  x.  I )  =  J ) )
4 divides 13558 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  ZZ )  ->  ( K  ||  L  <->  E. y  e.  ZZ  (
y  x.  K )  =  L ) )
53, 4bi2anan9 868 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  <->  ( E. x  e.  ZZ  ( x  x.  I )  =  J  /\  E. y  e.  ZZ  ( y  x.  K )  =  L ) ) )
76biimpd 207 . . . 4  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  ( E. x  e.  ZZ  (
x  x.  I )  =  J  /\  E. y  e.  ZZ  (
y  x.  K )  =  L ) ) )
8 reeanv 2909 . . . 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 227 . . 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 6119 . . . . . . 7  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
1312eqeq1d 2451 . . . . . 6  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
1413rspcev 3094 . . . . 5  |-  ( ( Z  e.  ZZ  /\  ( Z  x.  M
)  =  N )  ->  E. z  e.  ZZ  ( z  x.  M
)  =  N )
1510, 11, 14syl6an 545 . . . 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 2869 . . 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 13558 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2018, 19syl 16 . 2  |-  ( ph  ->  ( M  ||  N  <->  E. z  e.  ZZ  (
z  x.  M )  =  N ) )
2117, 20sylibrd 234 1  |-  ( ph  ->  ( ( I  ||  J  /\  K  ||  L
)  ->  M  ||  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737   class class class wbr 4313  (class class class)co 6112    x. cmul 9308   ZZcz 10667    || cdivides 13556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-iota 5402  df-fv 5447  df-ov 6115  df-dvds 13557
This theorem is referenced by:  dvds2ln  13584  dvds2add  13585  dvds2sub  13586  dvdstr  13587
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