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Theorem ecopoveq 7005
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 6090 . . . 4  |-  ( ( z  =  A  /\  u  =  D )  ->  ( z  .+  u
)  =  ( A 
.+  D ) )
2 oveq12 6090 . . . 4  |-  ( ( w  =  B  /\  v  =  C )  ->  ( w  .+  v
)  =  ( B 
.+  C ) )
31, 2eqeqan12d 2451 . . 3  |-  ( ( ( z  =  A  /\  u  =  D )  /\  ( w  =  B  /\  v  =  C ) )  -> 
( ( z  .+  u )  =  ( w  .+  v )  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
43an42s 801 . 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 4955 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    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   {copab 4265    X. cxp 4876  (class class class)co 6081
This theorem is referenced by:  ecopovsym  7006  ecopovtrn  7007  ecopover  7008  enqbreq  8796  enrbreq  8942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-iota 5418  df-fv 5462  df-ov 6084
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