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Definition df-enr 6809
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 6392 . 2 class ~R
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 cnp 6387 . . . . . . 7 class P
54, 4cxp 4343 . . . . . 6 class (P × P)
63, 5wcel 1393 . . . . 5 wff 𝑥 ∈ (P × P)
7 vy . . . . . . 7 setvar 𝑦
87cv 1242 . . . . . 6 class 𝑦
98, 5wcel 1393 . . . . 5 wff 𝑦 ∈ (P × P)
106, 9wa 97 . . . 4 wff (𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P))
11 vz . . . . . . . . . . . . 13 setvar 𝑧
1211cv 1242 . . . . . . . . . . . 12 class 𝑧
13 vw . . . . . . . . . . . . 13 setvar 𝑤
1413cv 1242 . . . . . . . . . . . 12 class 𝑤
1512, 14cop 3378 . . . . . . . . . . 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 3378 . . . . . . . . . . 11 class 𝑣, 𝑢
228, 21wceq 1243 . . . . . . . . . 10 wff 𝑦 = ⟨𝑣, 𝑢
2316, 22wa 97 . . . . . . . . 9 wff (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)
24 cpp 6389 . . . . . . . . . . 11 class +P
2512, 20, 24co 5512 . . . . . . . . . 10 class (𝑧 +P 𝑢)
2614, 18, 24co 5512 . . . . . . . . . 10 class (𝑤 +P 𝑣)
2725, 26wceq 1243 . . . . . . . . 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 1381 . . . . . . 7 wff 𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣))
3029, 17wex 1381 . . . . . 6 wff 𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣))
3130, 13wex 1381 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣))
3231, 11wex 1381 . . . 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 3817 . 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 1243 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  6817  enrer  6818  enrex  6820  prsrlem1  6825
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