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Theorem ecopoveq 6492
Description: This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation 
.~ (specified by the hypothesis) in terms of its operation  F. (Contributed by NM, 16-Aug-1995.)
Hypothesis
Ref Expression
ecopopr.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
Assertion
Ref Expression
ecopoveq  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .~  <. C ,  D >.  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
Distinct variable groups:    x, y, z, w, v, u,  .+    x, S, y, z, w, v, u    x, A, y, z, w, v, u    x, B, y, z, w, v, u   
x, C, y, z, w, v, u    x, D, y, z, w, v, u
Allowed substitution hints:    .~ ( x, y,
z, w, v, u)

Proof of Theorem ecopoveq
StepHypRef Expression
1 oveq12 5751 . . . 4  |-  ( ( z  =  A  /\  u  =  D )  ->  ( z  .+  u
)  =  ( A 
.+  D ) )
2 oveq12 5751 . . . 4  |-  ( ( w  =  B  /\  v  =  C )  ->  ( w  .+  v
)  =  ( B 
.+  C ) )
31, 2eqeqan12d 2133 . . 3  |-  ( ( ( z  =  A  /\  u  =  D )  /\  ( w  =  B  /\  v  =  C ) )  -> 
( ( z  .+  u )  =  ( w  .+  v )  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
43an42s 563 . 2  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ( v  =  C  /\  u  =  D ) )  -> 
( ( z  .+  u )  =  ( w  .+  v )  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
5 ecopopr.1 . 2  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z 
.+  u )  =  ( w  .+  v
) ) ) }
64, 5opbrop 4588 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  .~  <. C ,  D >.  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   <.cop 3500   class class class wbr 3899   {copab 3958    X. cxp 4507  (class class class)co 5742
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-xp 4515  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  ecopovsym  6493  ecopovtrn  6494  ecopover  6495  ecopovsymg  6496  ecopovtrng  6497  ecopoverg  6498  enqbreq  7132  enrbreq  7510  prsrlem1  7518
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