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Definition df-enq0 6579
Description: Define equivalence relation for non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
Assertion
Ref Expression
df-enq0  |- ~Q0  =  { <. x ,  y >.  |  ( ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) ) ) }
Distinct variable group:    x, y, z, w, v, u

Detailed syntax breakdown of Definition df-enq0
StepHypRef Expression
1 ceq0 6441 . 2  class ~Q0
2 vx . . . . . . 7  setvar  x
32cv 1258 . . . . . 6  class  x
4 com 4340 . . . . . . 7  class  om
5 cnpi 6427 . . . . . . 7  class  N.
64, 5cxp 4370 . . . . . 6  class  ( om 
X.  N. )
73, 6wcel 1409 . . . . 5  wff  x  e.  ( om  X.  N. )
8 vy . . . . . . 7  setvar  y
98cv 1258 . . . . . 6  class  y
109, 6wcel 1409 . . . . 5  wff  y  e.  ( om  X.  N. )
117, 10wa 101 . . . 4  wff  ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )
12 vz . . . . . . . . . . . . 13  setvar  z
1312cv 1258 . . . . . . . . . . . 12  class  z
14 vw . . . . . . . . . . . . 13  setvar  w
1514cv 1258 . . . . . . . . . . . 12  class  w
1613, 15cop 3405 . . . . . . . . . . 11  class  <. z ,  w >.
173, 16wceq 1259 . . . . . . . . . 10  wff  x  = 
<. z ,  w >.
18 vv . . . . . . . . . . . . 13  setvar  v
1918cv 1258 . . . . . . . . . . . 12  class  v
20 vu . . . . . . . . . . . . 13  setvar  u
2120cv 1258 . . . . . . . . . . . 12  class  u
2219, 21cop 3405 . . . . . . . . . . 11  class  <. v ,  u >.
239, 22wceq 1259 . . . . . . . . . 10  wff  y  = 
<. v ,  u >.
2417, 23wa 101 . . . . . . . . 9  wff  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
25 comu 6029 . . . . . . . . . . 11  class  .o
2613, 21, 25co 5539 . . . . . . . . . 10  class  ( z  .o  u )
2715, 19, 25co 5539 . . . . . . . . . 10  class  ( w  .o  v )
2826, 27wceq 1259 . . . . . . . . 9  wff  ( z  .o  u )  =  ( w  .o  v
)
2924, 28wa 101 . . . . . . . 8  wff  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u )  =  ( w  .o  v
) )
3029, 20wex 1397 . . . . . . 7  wff  E. u
( ( x  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  (
z  .o  u )  =  ( w  .o  v ) )
3130, 18wex 1397 . . . . . 6  wff  E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) )
3231, 14wex 1397 . . . . 5  wff  E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u )  =  ( w  .o  v
) )
3332, 12wex 1397 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) )
3411, 33wa 101 . . 3  wff  ( ( x  e.  ( om 
X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) ) )
3534, 2, 8copab 3844 . 2  class  { <. x ,  y >.  |  ( ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) ) ) }
361, 35wceq 1259 1  wff ~Q0  =  { <. x ,  y >.  |  ( ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u
)  =  ( w  .o  v ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  enq0enq  6586  enq0sym  6587  enq0ref  6588  enq0tr  6589  enq0er  6590  enq0breq  6591  enq0ex  6594
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