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Definition df-enq0 6406
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 6270 . 2 class ~Q0
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 com 4256 . . . . . . 7 class 𝜔
5 cnpi 6256 . . . . . . 7 class N
64, 5cxp 4286 . . . . . 6 class (𝜔 × N)
73, 6wcel 1390 . . . . 5 wff x (𝜔 × N)
8 vy . . . . . . 7 setvar y
98cv 1241 . . . . . 6 class y
109, 6wcel 1390 . . . . 5 wff y (𝜔 × N)
117, 10wa 97 . . . 4 wff (x (𝜔 × N) y (𝜔 × N))
12 vz . . . . . . . . . . . . 13 setvar z
1312cv 1241 . . . . . . . . . . . 12 class z
14 vw . . . . . . . . . . . . 13 setvar w
1514cv 1241 . . . . . . . . . . . 12 class w
1613, 15cop 3370 . . . . . . . . . . 11 class z, w
173, 16wceq 1242 . . . . . . . . . 10 wff x = ⟨z, w
18 vv . . . . . . . . . . . . 13 setvar v
1918cv 1241 . . . . . . . . . . . 12 class v
20 vu . . . . . . . . . . . . 13 setvar u
2120cv 1241 . . . . . . . . . . . 12 class u
2219, 21cop 3370 . . . . . . . . . . 11 class v, u
239, 22wceq 1242 . . . . . . . . . 10 wff y = ⟨v, u
2417, 23wa 97 . . . . . . . . 9 wff (x = ⟨z, w y = ⟨v, u⟩)
25 comu 5938 . . . . . . . . . . 11 class ·𝑜
2613, 21, 25co 5455 . . . . . . . . . 10 class (z ·𝑜 u)
2715, 19, 25co 5455 . . . . . . . . . 10 class (w ·𝑜 v)
2826, 27wceq 1242 . . . . . . . . 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 1378 . . . . . . 7 wff u((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))
3130, 18wex 1378 . . . . . 6 wff vu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))
3231, 14wex 1378 . . . . 5 wff wvu((x = ⟨z, w y = ⟨v, u⟩) (z ·𝑜 u) = (w ·𝑜 v))
3332, 12wex 1378 . . . 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 3808 . 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 1242 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  6413  enq0sym  6414  enq0ref  6415  enq0tr  6416  enq0er  6417  enq0breq  6418  enq0ex  6421
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