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Definition df-enr 6614
Description: Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
Assertion
Ref Expression
df-enr ~R = {⟨x, y⟩ ∣ ((x (P × P) y (P × P)) zwvu((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 6280 . 2 class ~R
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cnp 6275 . . . . . . 7 class P
54, 4cxp 4286 . . . . . 6 class (P × P)
63, 5wcel 1390 . . . . 5 wff x (P × P)
7 vy . . . . . . 7 setvar y
87cv 1241 . . . . . 6 class y
98, 5wcel 1390 . . . . 5 wff y (P × P)
106, 9wa 97 . . . 4 wff (x (P × P) y (P × P))
11 vz . . . . . . . . . . . . 13 setvar z
1211cv 1241 . . . . . . . . . . . 12 class z
13 vw . . . . . . . . . . . . 13 setvar w
1413cv 1241 . . . . . . . . . . . 12 class w
1512, 14cop 3370 . . . . . . . . . . 11 class z, w
163, 15wceq 1242 . . . . . . . . . 10 wff x = ⟨z, w
17 vv . . . . . . . . . . . . 13 setvar v
1817cv 1241 . . . . . . . . . . . 12 class v
19 vu . . . . . . . . . . . . 13 setvar u
2019cv 1241 . . . . . . . . . . . 12 class u
2118, 20cop 3370 . . . . . . . . . . 11 class v, u
228, 21wceq 1242 . . . . . . . . . 10 wff y = ⟨v, u
2316, 22wa 97 . . . . . . . . 9 wff (x = ⟨z, w y = ⟨v, u⟩)
24 cpp 6277 . . . . . . . . . . 11 class +P
2512, 20, 24co 5455 . . . . . . . . . 10 class (z +P u)
2614, 18, 24co 5455 . . . . . . . . . 10 class (w +P v)
2725, 26wceq 1242 . . . . . . . . 9 wff (z +P u) = (w +P v)
2823, 27wa 97 . . . . . . . 8 wff ((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v))
2928, 19wex 1378 . . . . . . 7 wff u((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v))
3029, 17wex 1378 . . . . . 6 wff vu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v))
3130, 13wex 1378 . . . . 5 wff wvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v))
3231, 11wex 1378 . . . 4 wff zwvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v))
3310, 32wa 97 . . 3 wff ((x (P × P) y (P × P)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v)))
3433, 2, 7copab 3808 . 2 class {⟨x, y⟩ ∣ ((x (P × P) y (P × P)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v)))}
351, 34wceq 1242 1 wff ~R = {⟨x, y⟩ ∣ ((x (P × P) y (P × P)) zwvu((x = ⟨z, w y = ⟨v, u⟩) (z +P u) = (w +P v)))}
Colors of variables: wff set class
This definition is referenced by:  enrbreq  6622  enrer  6623  enrex  6625  prsrlem1  6630
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