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Definition df-mqqs 6446
Description: Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
df-mqqs ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-mqqs
StepHypRef Expression
1 cmq 6379 . 2 class ·Q
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 cnq 6376 . . . . . 6 class Q
53, 4wcel 1393 . . . . 5 wff 𝑥Q
6 vy . . . . . . 7 setvar 𝑦
76cv 1242 . . . . . 6 class 𝑦
87, 4wcel 1393 . . . . 5 wff 𝑦Q
95, 8wa 97 . . . 4 wff (𝑥Q𝑦Q)
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 ceq 6375 . . . . . . . . . . . 12 class ~Q
1614, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑤, 𝑣⟩] ~Q
173, 16wceq 1243 . . . . . . . . . 10 wff 𝑥 = [⟨𝑤, 𝑣⟩] ~Q
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⟩] ~Q
247, 23wceq 1243 . . . . . . . . . 10 wff 𝑦 = [⟨𝑢, 𝑓⟩] ~Q
2517, 24wa 97 . . . . . . . . 9 wff (𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q )
26 vz . . . . . . . . . . 11 setvar 𝑧
2726cv 1242 . . . . . . . . . 10 class 𝑧
28 cmpq 6373 . . . . . . . . . . . 12 class ·pQ
2914, 22, 28co 5512 . . . . . . . . . . 11 class (⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)
3029, 15cec 6104 . . . . . . . . . 10 class [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q
3127, 30wceq 1243 . . . . . . . . 9 wff 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q
3225, 31wa 97 . . . . . . . 8 wff ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )
3332, 20wex 1381 . . . . . . 7 wff 𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )
3433, 18wex 1381 . . . . . 6 wff 𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )
3534, 12wex 1381 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )
3635, 10wex 1381 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q )
379, 36wa 97 . . 3 wff ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))
3837, 2, 6, 26coprab 5513 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
391, 38wceq 1243 1 wff ·Q = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q𝑦 = [⟨𝑢, 𝑓⟩] ~Q ) ∧ 𝑧 = [(⟨𝑤, 𝑣⟩ ·pQ𝑢, 𝑓⟩)] ~Q ))}
Colors of variables: wff set class
This definition is referenced by:  mulpipqqs  6469  dmmulpq  6476
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