MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvds2lem Structured version   Unicode version

Theorem dvds2lem 13846
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 13838 . . . . . . 7  |-  ( ( I  e.  ZZ  /\  J  e.  ZZ )  ->  ( I  ||  J  <->  E. x  e.  ZZ  (
x  x.  I )  =  J ) )
4 divides 13838 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  L  e.  ZZ )  ->  ( K  ||  L  <->  E. y  e.  ZZ  (
y  x.  K )  =  L ) )
53, 4bi2anan9 869 . . . . . 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 3022 . . . 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 6282 . . . . . . 7  |-  ( z  =  Z  ->  (
z  x.  M )  =  ( Z  x.  M ) )
1312eqeq1d 2462 . . . . . 6  |-  ( z  =  Z  ->  (
( z  x.  M
)  =  N  <->  ( Z  x.  M )  =  N ) )
1413rspcev 3207 . . . . 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 2955 . . 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 13838 . . 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 1374    e. wcel 1762   E.wrex 2808   class class class wbr 4440  (class class class)co 6275    x. cmul 9486   ZZcz 10853    || cdivides 13836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-iota 5542  df-fv 5587  df-ov 6278  df-dvds 13837
This theorem is referenced by:  dvds2ln  13864  dvds2add  13865  dvds2sub  13866  dvdstr  13867
  Copyright terms: Public domain W3C validator