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Definition df-enr 9330
Description: Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9392, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-enr  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
Distinct variable group:    x, y, z, w, v, u

Detailed syntax breakdown of Definition df-enr
StepHypRef Expression
1 cer 9137 . 2  class  ~R
2 vx . . . . . . 7  setvar  x
32cv 1369 . . . . . 6  class  x
4 cnp 9130 . . . . . . 7  class  P.
54, 4cxp 4939 . . . . . 6  class  ( P. 
X.  P. )
63, 5wcel 1758 . . . . 5  wff  x  e.  ( P.  X.  P. )
7 vy . . . . . . 7  setvar  y
87cv 1369 . . . . . 6  class  y
98, 5wcel 1758 . . . . 5  wff  y  e.  ( P.  X.  P. )
106, 9wa 369 . . . 4  wff  ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. ) )
11 vz . . . . . . . . . . . . 13  setvar  z
1211cv 1369 . . . . . . . . . . . 12  class  z
13 vw . . . . . . . . . . . . 13  setvar  w
1413cv 1369 . . . . . . . . . . . 12  class  w
1512, 14cop 3984 . . . . . . . . . . 11  class  <. z ,  w >.
163, 15wceq 1370 . . . . . . . . . 10  wff  x  = 
<. z ,  w >.
17 vv . . . . . . . . . . . . 13  setvar  v
1817cv 1369 . . . . . . . . . . . 12  class  v
19 vu . . . . . . . . . . . . 13  setvar  u
2019cv 1369 . . . . . . . . . . . 12  class  u
2118, 20cop 3984 . . . . . . . . . . 11  class  <. v ,  u >.
228, 21wceq 1370 . . . . . . . . . 10  wff  y  = 
<. v ,  u >.
2316, 22wa 369 . . . . . . . . 9  wff  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
24 cpp 9132 . . . . . . . . . . 11  class  +P.
2512, 20, 24co 6193 . . . . . . . . . 10  class  ( z  +P.  u )
2614, 18, 24co 6193 . . . . . . . . . 10  class  ( w  +P.  v )
2725, 26wceq 1370 . . . . . . . . 9  wff  ( z  +P.  u )  =  ( w  +P.  v
)
2823, 27wa 369 . . . . . . . 8  wff  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u )  =  ( w  +P.  v
) )
2928, 19wex 1587 . . . . . . 7  wff  E. u
( ( x  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  (
z  +P.  u )  =  ( w  +P.  v ) )
3029, 17wex 1587 . . . . . 6  wff  E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) )
3130, 13wex 1587 . . . . 5  wff  E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u )  =  ( w  +P.  v
) )
3231, 11wex 1587 . . . 4  wff  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) )
3310, 32wa 369 . . 3  wff  ( ( x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) )
3433, 2, 7copab 4450 . 2  class  { <. x ,  y >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
351, 34wceq 1370 1  wff  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  +P.  u
)  =  ( w  +P.  v ) ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  enrbreq  9338  enrer  9339  enrex  9341  addsrpr  9346  mulsrpr  9347
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