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Definition df-plq0 6523
Description: Define addition on 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-plq0 +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-plq0
StepHypRef Expression
1 cplq0 6385 . 2 class +Q0
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 cnq0 6383 . . . . . 6 class Q0
53, 4wcel 1393 . . . . 5 wff 𝑥Q0
6 vy . . . . . . 7 setvar 𝑦
76cv 1242 . . . . . 6 class 𝑦
87, 4wcel 1393 . . . . 5 wff 𝑦Q0
95, 8wa 97 . . . 4 wff (𝑥Q0𝑦Q0)
10 vw . . . . . . . . . . . . . 14 setvar 𝑤
1110cv 1242 . . . . . . . . . . . . 13 class 𝑤
12 vv . . . . . . . . . . . . . 14 setvar 𝑣
1312cv 1242 . . . . . . . . . . . . 13 class 𝑣
1411, 13cop 3378 . . . . . . . . . . . 12 class 𝑤, 𝑣
15 ceq0 6382 . . . . . . . . . . . 12 class ~Q0
1614, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑤, 𝑣⟩] ~Q0
173, 16wceq 1243 . . . . . . . . . 10 wff 𝑥 = [⟨𝑤, 𝑣⟩] ~Q0
18 vu . . . . . . . . . . . . . 14 setvar 𝑢
1918cv 1242 . . . . . . . . . . . . 13 class 𝑢
20 vf . . . . . . . . . . . . . 14 setvar 𝑓
2120cv 1242 . . . . . . . . . . . . 13 class 𝑓
2219, 21cop 3378 . . . . . . . . . . . 12 class 𝑢, 𝑓
2322, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑢, 𝑓⟩] ~Q0
247, 23wceq 1243 . . . . . . . . . 10 wff 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0
2517, 24wa 97 . . . . . . . . 9 wff (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 )
26 vz . . . . . . . . . . 11 setvar 𝑧
2726cv 1242 . . . . . . . . . 10 class 𝑧
28 comu 5999 . . . . . . . . . . . . . 14 class ·𝑜
2911, 21, 28co 5512 . . . . . . . . . . . . 13 class (𝑤 ·𝑜 𝑓)
3013, 19, 28co 5512 . . . . . . . . . . . . 13 class (𝑣 ·𝑜 𝑢)
31 coa 5998 . . . . . . . . . . . . 13 class +𝑜
3229, 30, 31co 5512 . . . . . . . . . . . 12 class ((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢))
3313, 21, 28co 5512 . . . . . . . . . . . 12 class (𝑣 ·𝑜 𝑓)
3432, 33cop 3378 . . . . . . . . . . 11 class ⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩
3534, 15cec 6104 . . . . . . . . . 10 class [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0
3627, 35wceq 1243 . . . . . . . . 9 wff 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0
3725, 36wa 97 . . . . . . . 8 wff ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 )
3837, 20wex 1381 . . . . . . 7 wff 𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 )
3938, 18wex 1381 . . . . . 6 wff 𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 )
4039, 12wex 1381 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 )
4140, 10wex 1381 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 )
429, 41wa 97 . . 3 wff ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))
4342, 2, 6, 26coprab 5513 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}
441, 43wceq 1243 1 wff +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q0𝑦Q0) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}
Colors of variables: wff set class
This definition is referenced by:  dfplq0qs  6526
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