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Definition df-enq0 6520
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 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((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 6382 . 2 class ~Q0
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 com 4313 . . . . . . 7 class ω
5 cnpi 6368 . . . . . . 7 class N
64, 5cxp 4343 . . . . . 6 class (ω × N)
73, 6wcel 1393 . . . . 5 wff 𝑥 ∈ (ω × N)
8 vy . . . . . . 7 setvar 𝑦
98cv 1242 . . . . . 6 class 𝑦
109, 6wcel 1393 . . . . 5 wff 𝑦 ∈ (ω × N)
117, 10wa 97 . . . 4 wff (𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))
12 vz . . . . . . . . . . . . 13 setvar 𝑧
1312cv 1242 . . . . . . . . . . . 12 class 𝑧
14 vw . . . . . . . . . . . . 13 setvar 𝑤
1514cv 1242 . . . . . . . . . . . 12 class 𝑤
1613, 15cop 3378 . . . . . . . . . . 11 class 𝑧, 𝑤
173, 16wceq 1243 . . . . . . . . . 10 wff 𝑥 = ⟨𝑧, 𝑤
18 vv . . . . . . . . . . . . 13 setvar 𝑣
1918cv 1242 . . . . . . . . . . . 12 class 𝑣
20 vu . . . . . . . . . . . . 13 setvar 𝑢
2120cv 1242 . . . . . . . . . . . 12 class 𝑢
2219, 21cop 3378 . . . . . . . . . . 11 class 𝑣, 𝑢
239, 22wceq 1243 . . . . . . . . . 10 wff 𝑦 = ⟨𝑣, 𝑢
2417, 23wa 97 . . . . . . . . 9 wff (𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)
25 comu 5999 . . . . . . . . . . 11 class ·𝑜
2613, 21, 25co 5512 . . . . . . . . . 10 class (𝑧 ·𝑜 𝑢)
2715, 19, 25co 5512 . . . . . . . . . 10 class (𝑤 ·𝑜 𝑣)
2826, 27wceq 1243 . . . . . . . . 9 wff (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)
2924, 28wa 97 . . . . . . . 8 wff ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
3029, 20wex 1381 . . . . . . 7 wff 𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
3130, 18wex 1381 . . . . . 6 wff 𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
3231, 14wex 1381 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
3332, 12wex 1381 . . . 4 wff 𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
3411, 33wa 97 . . 3 wff ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))
3534, 2, 8copab 3817 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))}
361, 35wceq 1243 1 wff ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))}
Colors of variables: wff set class
This definition is referenced by:  enq0enq  6527  enq0sym  6528  enq0ref  6529  enq0tr  6530  enq0er  6531  enq0breq  6532  enq0ex  6535
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