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Definition df-ltr 8653
Description: Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
df-ltr  |-  <R  =  { <. x ,  y
>.  |  ( (
x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
Distinct variable group:    x, y, z, w, v, u

Detailed syntax breakdown of Definition df-ltr
StepHypRef Expression
1 cltr 8463 . 2  class  <R
2 vx . . . . . . 7  set  x
32cv 1618 . . . . . 6  class  x
4 cnr 8457 . . . . . 6  class  R.
53, 4wcel 1621 . . . . 5  wff  x  e. 
R.
6 vy . . . . . . 7  set  y
76cv 1618 . . . . . 6  class  y
87, 4wcel 1621 . . . . 5  wff  y  e. 
R.
95, 8wa 360 . . . 4  wff  ( x  e.  R.  /\  y  e.  R. )
10 vz . . . . . . . . . . . . . 14  set  z
1110cv 1618 . . . . . . . . . . . . 13  class  z
12 vw . . . . . . . . . . . . . 14  set  w
1312cv 1618 . . . . . . . . . . . . 13  class  w
1411, 13cop 3617 . . . . . . . . . . . 12  class  <. z ,  w >.
15 cer 8456 . . . . . . . . . . . 12  class  ~R
1614, 15cec 6626 . . . . . . . . . . 11  class  [ <. z ,  w >. ]  ~R
173, 16wceq 1619 . . . . . . . . . 10  wff  x  =  [ <. z ,  w >. ]  ~R
18 vv . . . . . . . . . . . . . 14  set  v
1918cv 1618 . . . . . . . . . . . . 13  class  v
20 vu . . . . . . . . . . . . . 14  set  u
2120cv 1618 . . . . . . . . . . . . 13  class  u
2219, 21cop 3617 . . . . . . . . . . . 12  class  <. v ,  u >.
2322, 15cec 6626 . . . . . . . . . . 11  class  [ <. v ,  u >. ]  ~R
247, 23wceq 1619 . . . . . . . . . 10  wff  y  =  [ <. v ,  u >. ]  ~R
2517, 24wa 360 . . . . . . . . 9  wff  ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )
26 cpp 8451 . . . . . . . . . . 11  class  +P.
2711, 21, 26co 5792 . . . . . . . . . 10  class  ( z  +P.  u )
2813, 19, 26co 5792 . . . . . . . . . 10  class  ( w  +P.  v )
29 cltp 8453 . . . . . . . . . 10  class  <P
3027, 28, 29wbr 3997 . . . . . . . . 9  wff  ( z  +P.  u )  <P 
( w  +P.  v
)
3125, 30wa 360 . . . . . . . 8  wff  ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u )  <P  (
w  +P.  v )
)
3231, 20wex 1537 . . . . . . 7  wff  E. u
( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) )
3332, 18wex 1537 . . . . . 6  wff  E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) )
3433, 12wex 1537 . . . . 5  wff  E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) )
3534, 10wex 1537 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) )
369, 35wa 360 . . 3  wff  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) )
3736, 2, 6copab 4050 . 2  class  { <. x ,  y >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
381, 37wceq 1619 1  wff  <R  =  { <. x ,  y
>.  |  ( (
x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  ltrelsr  8661  ltsrpr  8667
  Copyright terms: Public domain W3C validator