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Definition df-ltr 9919
Description: Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9980, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
df-ltr <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Detailed syntax breakdown of Definition df-ltr
StepHypRef Expression
1 cltr 9731 . 2 class <R
2 vx . . . . . . 7 setvar 𝑥
32cv 1522 . . . . . 6 class 𝑥
4 cnr 9725 . . . . . 6 class R
53, 4wcel 2030 . . . . 5 wff 𝑥R
6 vy . . . . . . 7 setvar 𝑦
76cv 1522 . . . . . 6 class 𝑦
87, 4wcel 2030 . . . . 5 wff 𝑦R
95, 8wa 383 . . . 4 wff (𝑥R𝑦R)
10 vz . . . . . . . . . . . . . 14 setvar 𝑧
1110cv 1522 . . . . . . . . . . . . 13 class 𝑧
12 vw . . . . . . . . . . . . . 14 setvar 𝑤
1312cv 1522 . . . . . . . . . . . . 13 class 𝑤
1411, 13cop 4216 . . . . . . . . . . . 12 class 𝑧, 𝑤
15 cer 9724 . . . . . . . . . . . 12 class ~R
1614, 15cec 7785 . . . . . . . . . . 11 class [⟨𝑧, 𝑤⟩] ~R
173, 16wceq 1523 . . . . . . . . . 10 wff 𝑥 = [⟨𝑧, 𝑤⟩] ~R
18 vv . . . . . . . . . . . . . 14 setvar 𝑣
1918cv 1522 . . . . . . . . . . . . 13 class 𝑣
20 vu . . . . . . . . . . . . . 14 setvar 𝑢
2120cv 1522 . . . . . . . . . . . . 13 class 𝑢
2219, 21cop 4216 . . . . . . . . . . . 12 class 𝑣, 𝑢
2322, 15cec 7785 . . . . . . . . . . 11 class [⟨𝑣, 𝑢⟩] ~R
247, 23wceq 1523 . . . . . . . . . 10 wff 𝑦 = [⟨𝑣, 𝑢⟩] ~R
2517, 24wa 383 . . . . . . . . 9 wff (𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R )
26 cpp 9721 . . . . . . . . . . 11 class +P
2711, 21, 26co 6690 . . . . . . . . . 10 class (𝑧 +P 𝑢)
2813, 19, 26co 6690 . . . . . . . . . 10 class (𝑤 +P 𝑣)
29 cltp 9723 . . . . . . . . . 10 class <P
3027, 28, 29wbr 4685 . . . . . . . . 9 wff (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)
3125, 30wa 383 . . . . . . . 8 wff ((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3231, 20wex 1744 . . . . . . 7 wff 𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3332, 18wex 1744 . . . . . 6 wff 𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3433, 12wex 1744 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3534, 10wex 1744 . . . 4 wff 𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
369, 35wa 383 . . 3 wff ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
3736, 2, 6copab 4745 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
381, 37wceq 1523 1 wff <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
Colors of variables: wff setvar class
This definition is referenced by:  ltrelsr  9927  ltsrpr  9936
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