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Theorem ecopoveq 6759
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 5867 . . . 4  |-  ( ( z  =  A  /\  u  =  D )  ->  ( z  .+  u
)  =  ( A 
.+  D ) )
2 oveq12 5867 . . . 4  |-  ( ( w  =  B  /\  v  =  C )  ->  ( w  .+  v
)  =  ( B 
.+  C ) )
31, 2eqeqan12d 2298 . . 3  |-  ( ( ( z  =  A  /\  u  =  D )  /\  ( w  =  B  /\  v  =  C ) )  -> 
( ( z  .+  u )  =  ( w  .+  v )  <-> 
( A  .+  D
)  =  ( B 
.+  C ) ) )
43an42s 800 . 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 4767 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687  (class class class)co 5858
This theorem is referenced by:  ecopovsym  6760  ecopovtrn  6761  ecopover  6762  enqbreq  8543  enrbreq  8689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-iota 5219  df-fv 5263  df-ov 5861
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