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Definition df-enq 9693
Description: Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 9902, 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.) (New usage is discouraged.)
Assertion
Ref Expression
df-enq ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((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 9633 . 2 class ~Q
2 vx . . . . . . 7 setvar 𝑥
32cv 1479 . . . . . 6 class 𝑥
4 cnpi 9626 . . . . . . 7 class N
54, 4cxp 5082 . . . . . 6 class (N × N)
63, 5wcel 1987 . . . . 5 wff 𝑥 ∈ (N × N)
7 vy . . . . . . 7 setvar 𝑦
87cv 1479 . . . . . 6 class 𝑦
98, 5wcel 1987 . . . . 5 wff 𝑦 ∈ (N × N)
106, 9wa 384 . . . 4 wff (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))
11 vz . . . . . . . . . . . . 13 setvar 𝑧
1211cv 1479 . . . . . . . . . . . 12 class 𝑧
13 vw . . . . . . . . . . . . 13 setvar 𝑤
1413cv 1479 . . . . . . . . . . . 12 class 𝑤
1512, 14cop 4161 . . . . . . . . . . 11 class 𝑧, 𝑤
163, 15wceq 1480 . . . . . . . . . 10 wff 𝑥 = ⟨𝑧, 𝑤
17 vv . . . . . . . . . . . . 13 setvar 𝑣
1817cv 1479 . . . . . . . . . . . 12 class 𝑣
19 vu . . . . . . . . . . . . 13 setvar 𝑢
2019cv 1479 . . . . . . . . . . . 12 class 𝑢
2118, 20cop 4161 . . . . . . . . . . 11 class 𝑣, 𝑢
228, 21wceq 1480 . . . . . . . . . 10 wff 𝑦 = ⟨𝑣, 𝑢
2316, 22wa 384 . . . . . . . . 9 wff (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)
24 cmi 9628 . . . . . . . . . . 11 class ·N
2512, 20, 24co 6615 . . . . . . . . . 10 class (𝑧 ·N 𝑢)
2614, 18, 24co 6615 . . . . . . . . . 10 class (𝑤 ·N 𝑣)
2725, 26wceq 1480 . . . . . . . . 9 wff (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)
2823, 27wa 384 . . . . . . . 8 wff ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))
2928, 19wex 1701 . . . . . . 7 wff 𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))
3029, 17wex 1701 . . . . . 6 wff 𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))
3130, 13wex 1701 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))
3231, 11wex 1701 . . . 4 wff 𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣))
3310, 32wa 384 . . 3 wff ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))
3433, 2, 7copab 4682 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
351, 34wceq 1480 1 wff ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
Colors of variables: wff setvar class
This definition is referenced by:  enqbreq  9701  enqer  9703  enqex  9704
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