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Definition df-mr 9736
Description: Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9798, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-mr ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-mr
StepHypRef Expression
1 cmr 9548 . 2 class ·R
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 cnr 9543 . . . . . 6 class R
53, 4wcel 1976 . . . . 5 wff 𝑥R
6 vy . . . . . . 7 setvar 𝑦
76cv 1473 . . . . . 6 class 𝑦
87, 4wcel 1976 . . . . 5 wff 𝑦R
95, 8wa 382 . . . 4 wff (𝑥R𝑦R)
10 vw . . . . . . . . . . . . . 14 setvar 𝑤
1110cv 1473 . . . . . . . . . . . . 13 class 𝑤
12 vv . . . . . . . . . . . . . 14 setvar 𝑣
1312cv 1473 . . . . . . . . . . . . 13 class 𝑣
1411, 13cop 4130 . . . . . . . . . . . 12 class 𝑤, 𝑣
15 cer 9542 . . . . . . . . . . . 12 class ~R
1614, 15cec 7604 . . . . . . . . . . 11 class [⟨𝑤, 𝑣⟩] ~R
173, 16wceq 1474 . . . . . . . . . 10 wff 𝑥 = [⟨𝑤, 𝑣⟩] ~R
18 vu . . . . . . . . . . . . . 14 setvar 𝑢
1918cv 1473 . . . . . . . . . . . . 13 class 𝑢
20 vf . . . . . . . . . . . . . 14 setvar 𝑓
2120cv 1473 . . . . . . . . . . . . 13 class 𝑓
2219, 21cop 4130 . . . . . . . . . . . 12 class 𝑢, 𝑓
2322, 15cec 7604 . . . . . . . . . . 11 class [⟨𝑢, 𝑓⟩] ~R
247, 23wceq 1474 . . . . . . . . . 10 wff 𝑦 = [⟨𝑢, 𝑓⟩] ~R
2517, 24wa 382 . . . . . . . . 9 wff (𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R )
26 vz . . . . . . . . . . 11 setvar 𝑧
2726cv 1473 . . . . . . . . . 10 class 𝑧
28 cmp 9540 . . . . . . . . . . . . . 14 class ·P
2911, 19, 28co 6527 . . . . . . . . . . . . 13 class (𝑤 ·P 𝑢)
3013, 21, 28co 6527 . . . . . . . . . . . . 13 class (𝑣 ·P 𝑓)
31 cpp 9539 . . . . . . . . . . . . 13 class +P
3229, 30, 31co 6527 . . . . . . . . . . . 12 class ((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓))
3311, 21, 28co 6527 . . . . . . . . . . . . 13 class (𝑤 ·P 𝑓)
3413, 19, 28co 6527 . . . . . . . . . . . . 13 class (𝑣 ·P 𝑢)
3533, 34, 31co 6527 . . . . . . . . . . . 12 class ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))
3632, 35cop 4130 . . . . . . . . . . 11 class ⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩
3736, 15cec 7604 . . . . . . . . . 10 class [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R
3827, 37wceq 1474 . . . . . . . . 9 wff 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R
3925, 38wa 382 . . . . . . . 8 wff ((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R )
4039, 20wex 1694 . . . . . . 7 wff 𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R )
4140, 18wex 1694 . . . . . 6 wff 𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R )
4241, 12wex 1694 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R )
4342, 10wex 1694 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R )
449, 43wa 382 . . 3 wff ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))
4544, 2, 6, 26coprab 6528 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
461, 45wceq 1474 1 wff ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
Colors of variables: wff setvar class
This definition is referenced by:  mulsrpr  9753  dmmulsr  9763
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