ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-ltnqqs Unicode version

Definition df-ltnqqs 6423
Description: Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.)
Assertion
Ref Expression
df-ltnqqs  |-  <Q  =  { <. x ,  y
>.  |  ( (
x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) ) ) }
Distinct variable group:    x, y, z, w, v, u

Detailed syntax breakdown of Definition df-ltnqqs
StepHypRef Expression
1 cltq 6355 . 2  class  <Q
2 vx . . . . . . 7  setvar  x
32cv 1242 . . . . . 6  class  x
4 cnq 6350 . . . . . 6  class  Q.
53, 4wcel 1393 . . . . 5  wff  x  e. 
Q.
6 vy . . . . . . 7  setvar  y
76cv 1242 . . . . . 6  class  y
87, 4wcel 1393 . . . . 5  wff  y  e. 
Q.
95, 8wa 97 . . . 4  wff  ( x  e.  Q.  /\  y  e.  Q. )
10 vz . . . . . . . . . . . . . 14  setvar  z
1110cv 1242 . . . . . . . . . . . . 13  class  z
12 vw . . . . . . . . . . . . . 14  setvar  w
1312cv 1242 . . . . . . . . . . . . 13  class  w
1411, 13cop 3375 . . . . . . . . . . . 12  class  <. z ,  w >.
15 ceq 6349 . . . . . . . . . . . 12  class  ~Q
1614, 15cec 6082 . . . . . . . . . . 11  class  [ <. z ,  w >. ]  ~Q
173, 16wceq 1243 . . . . . . . . . 10  wff  x  =  [ <. z ,  w >. ]  ~Q
18 vv . . . . . . . . . . . . . 14  setvar  v
1918cv 1242 . . . . . . . . . . . . 13  class  v
20 vu . . . . . . . . . . . . . 14  setvar  u
2120cv 1242 . . . . . . . . . . . . 13  class  u
2219, 21cop 3375 . . . . . . . . . . . 12  class  <. v ,  u >.
2322, 15cec 6082 . . . . . . . . . . 11  class  [ <. v ,  u >. ]  ~Q
247, 23wceq 1243 . . . . . . . . . 10  wff  y  =  [ <. v ,  u >. ]  ~Q
2517, 24wa 97 . . . . . . . . 9  wff  ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )
26 cmi 6344 . . . . . . . . . . 11  class  .N
2711, 21, 26co 5490 . . . . . . . . . 10  class  ( z  .N  u )
2813, 19, 26co 5490 . . . . . . . . . 10  class  ( w  .N  v )
29 clti 6345 . . . . . . . . . 10  class  <N
3027, 28, 29wbr 3761 . . . . . . . . 9  wff  ( z  .N  u )  <N 
( w  .N  v
)
3125, 30wa 97 . . . . . . . 8  wff  ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u )  <N  (
w  .N  v ) )
3231, 20wex 1381 . . . . . . 7  wff  E. u
( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) )
3332, 18wex 1381 . . . . . 6  wff  E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) )
3433, 12wex 1381 . . . . 5  wff  E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) )
3534, 10wex 1381 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) )
369, 35wa 97 . . 3  wff  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) ) )
3736, 2, 6copab 3814 . 2  class  { <. x ,  y >.  |  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) ) ) }
381, 37wceq 1243 1  wff  <Q  =  { <. x ,  y
>.  |  ( (
x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u
)  <N  ( w  .N  v ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  ltrelnq  6435  ordpipqqs  6444
  Copyright terms: Public domain W3C validator