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Theorem ecopoveq 6964
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 6049 . . . 4  |-  ( ( z  =  A  /\  u  =  D )  ->  ( z  .+  u
)  =  ( A 
.+  D ) )
2 oveq12 6049 . . . 4  |-  ( ( w  =  B  /\  v  =  C )  ->  ( w  .+  v
)  =  ( B 
.+  C ) )
31, 2eqeqan12d 2419 . . 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 4914 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 1547    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172   {copab 4225    X. cxp 4835  (class class class)co 6040
This theorem is referenced by:  ecopovsym  6965  ecopovtrn  6966  ecopover  6967  enqbreq  8752  enrbreq  8898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-iota 5377  df-fv 5421  df-ov 6043
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