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Definition df-enq 8468
Description: Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8676, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-enq  |-  ~Q  =  { <. x ,  y
>.  |  ( (
x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) ) }
Distinct variable group:    x, y, z, w, v, u

Detailed syntax breakdown of Definition df-enq
StepHypRef Expression
1 ceq 8406 . 2  class  ~Q
2 vx . . . . . . 7  set  x
32cv 1618 . . . . . 6  class  x
4 cnpi 8399 . . . . . . 7  class  N.
54, 4cxp 4624 . . . . . 6  class  ( N. 
X.  N. )
63, 5wcel 1621 . . . . 5  wff  x  e.  ( N.  X.  N. )
7 vy . . . . . . 7  set  y
87cv 1618 . . . . . 6  class  y
98, 5wcel 1621 . . . . 5  wff  y  e.  ( N.  X.  N. )
106, 9wa 360 . . . 4  wff  ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )
11 vz . . . . . . . . . . . . 13  set  z
1211cv 1618 . . . . . . . . . . . 12  class  z
13 vw . . . . . . . . . . . . 13  set  w
1413cv 1618 . . . . . . . . . . . 12  class  w
1512, 14cop 3584 . . . . . . . . . . 11  class  <. z ,  w >.
163, 15wceq 1619 . . . . . . . . . 10  wff  x  = 
<. z ,  w >.
17 vv . . . . . . . . . . . . 13  set  v
1817cv 1618 . . . . . . . . . . . 12  class  v
19 vu . . . . . . . . . . . . 13  set  u
2019cv 1618 . . . . . . . . . . . 12  class  u
2118, 20cop 3584 . . . . . . . . . . 11  class  <. v ,  u >.
228, 21wceq 1619 . . . . . . . . . 10  wff  y  = 
<. v ,  u >.
2316, 22wa 360 . . . . . . . . 9  wff  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
24 cmi 8401 . . . . . . . . . . 11  class  .N
2512, 20, 24co 5757 . . . . . . . . . 10  class  ( z  .N  u )
2614, 18, 24co 5757 . . . . . . . . . 10  class  ( w  .N  v )
2725, 26wceq 1619 . . . . . . . . 9  wff  ( z  .N  u )  =  ( w  .N  v
)
2823, 27wa 360 . . . . . . . 8  wff  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u )  =  ( w  .N  v
) )
2928, 19wex 1537 . . . . . . 7  wff  E. u
( ( x  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  (
z  .N  u )  =  ( w  .N  v ) )
3029, 17wex 1537 . . . . . 6  wff  E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) )
3130, 13wex 1537 . . . . 5  wff  E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u )  =  ( w  .N  v
) )
3231, 11wex 1537 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) )
3310, 32wa 360 . . 3  wff  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) )
3433, 2, 7copab 4016 . 2  class  { <. x ,  y >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) ) }
351, 34wceq 1619 1  wff  ~Q  =  { <. x ,  y
>.  |  ( (
x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  enqbreq  8476  enqer  8478  enqex  8479
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