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Definition df-enq 6417
Description: Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, 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.)
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 6349 . 2  class  ~Q
2 vx . . . . . . 7  setvar  x
32cv 1242 . . . . . 6  class  x
4 cnpi 6342 . . . . . . 7  class  N.
54, 4cxp 4321 . . . . . 6  class  ( N. 
X.  N. )
63, 5wcel 1393 . . . . 5  wff  x  e.  ( N.  X.  N. )
7 vy . . . . . . 7  setvar  y
87cv 1242 . . . . . 6  class  y
98, 5wcel 1393 . . . . 5  wff  y  e.  ( N.  X.  N. )
106, 9wa 97 . . . 4  wff  ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )
11 vz . . . . . . . . . . . . 13  setvar  z
1211cv 1242 . . . . . . . . . . . 12  class  z
13 vw . . . . . . . . . . . . 13  setvar  w
1413cv 1242 . . . . . . . . . . . 12  class  w
1512, 14cop 3375 . . . . . . . . . . 11  class  <. z ,  w >.
163, 15wceq 1243 . . . . . . . . . 10  wff  x  = 
<. z ,  w >.
17 vv . . . . . . . . . . . . 13  setvar  v
1817cv 1242 . . . . . . . . . . . 12  class  v
19 vu . . . . . . . . . . . . 13  setvar  u
2019cv 1242 . . . . . . . . . . . 12  class  u
2118, 20cop 3375 . . . . . . . . . . 11  class  <. v ,  u >.
228, 21wceq 1243 . . . . . . . . . 10  wff  y  = 
<. v ,  u >.
2316, 22wa 97 . . . . . . . . 9  wff  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
24 cmi 6344 . . . . . . . . . . 11  class  .N
2512, 20, 24co 5490 . . . . . . . . . 10  class  ( z  .N  u )
2614, 18, 24co 5490 . . . . . . . . . 10  class  ( w  .N  v )
2725, 26wceq 1243 . . . . . . . . 9  wff  ( z  .N  u )  =  ( w  .N  v
)
2823, 27wa 97 . . . . . . . 8  wff  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u )  =  ( w  .N  v
) )
2928, 19wex 1381 . . . . . . 7  wff  E. u
( ( x  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  (
z  .N  u )  =  ( w  .N  v ) )
3029, 17wex 1381 . . . . . 6  wff  E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) )
3130, 13wex 1381 . . . . 5  wff  E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u )  =  ( w  .N  v
) )
3231, 11wex 1381 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .N  u
)  =  ( w  .N  v ) )
3310, 32wa 97 . . 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 3814 . 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 1243 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  6426  enqer  6428  enqex  6430  addpipqqs  6440  mulpipqqs  6443  enq0enq  6501
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