MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-plr Structured version   Visualization version   GIF version

Definition df-plr 9736
Description: Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9799, 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-plr +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓

Detailed syntax breakdown of Definition df-plr
StepHypRef Expression
1 cplr 9548 . 2 class +R
2 vx . . . . . . 7 setvar 𝑥
32cv 1473 . . . . . 6 class 𝑥
4 cnr 9544 . . . . . 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 9543 . . . . . . . . . . . 12 class ~R
1614, 15cec 7605 . . . . . . . . . . 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 7605 . . . . . . . . . . 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 cpp 9540 . . . . . . . . . . . . 13 class +P
2911, 19, 28co 6527 . . . . . . . . . . . 12 class (𝑤 +P 𝑢)
3013, 21, 28co 6527 . . . . . . . . . . . 12 class (𝑣 +P 𝑓)
3129, 30cop 4130 . . . . . . . . . . 11 class ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩
3231, 15cec 7605 . . . . . . . . . 10 class [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R
3327, 32wceq 1474 . . . . . . . . 9 wff 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R
3425, 33wa 382 . . . . . . . 8 wff ((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R )
3534, 20wex 1694 . . . . . . 7 wff 𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R )
3635, 18wex 1694 . . . . . 6 wff 𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R )
3736, 12wex 1694 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R )
3837, 10wex 1694 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R )
399, 38wa 382 . . 3 wff ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))
4039, 2, 6, 26coprab 6528 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
411, 40wceq 1474 1 wff +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
Colors of variables: wff setvar class
This definition is referenced by:  addsrpr  9753  dmaddsr  9763
  Copyright terms: Public domain W3C validator