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Definition df-ltr 6813
Description: Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)
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 6399 . 2 class <R
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 cnr 6393 . . . . . 6 class R
53, 4wcel 1393 . . . . 5 wff 𝑥R
6 vy . . . . . . 7 setvar 𝑦
76cv 1242 . . . . . 6 class 𝑦
87, 4wcel 1393 . . . . 5 wff 𝑦R
95, 8wa 97 . . . 4 wff (𝑥R𝑦R)
10 vz . . . . . . . . . . . . . 14 setvar 𝑧
1110cv 1242 . . . . . . . . . . . . 13 class 𝑧
12 vw . . . . . . . . . . . . . 14 setvar 𝑤
1312cv 1242 . . . . . . . . . . . . 13 class 𝑤
1411, 13cop 3378 . . . . . . . . . . . 12 class 𝑧, 𝑤
15 cer 6392 . . . . . . . . . . . 12 class ~R
1614, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑧, 𝑤⟩] ~R
173, 16wceq 1243 . . . . . . . . . 10 wff 𝑥 = [⟨𝑧, 𝑤⟩] ~R
18 vv . . . . . . . . . . . . . 14 setvar 𝑣
1918cv 1242 . . . . . . . . . . . . 13 class 𝑣
20 vu . . . . . . . . . . . . . 14 setvar 𝑢
2120cv 1242 . . . . . . . . . . . . 13 class 𝑢
2219, 21cop 3378 . . . . . . . . . . . 12 class 𝑣, 𝑢
2322, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑣, 𝑢⟩] ~R
247, 23wceq 1243 . . . . . . . . . 10 wff 𝑦 = [⟨𝑣, 𝑢⟩] ~R
2517, 24wa 97 . . . . . . . . 9 wff (𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R )
26 cpp 6389 . . . . . . . . . . 11 class +P
2711, 21, 26co 5512 . . . . . . . . . 10 class (𝑧 +P 𝑢)
2813, 19, 26co 5512 . . . . . . . . . 10 class (𝑤 +P 𝑣)
29 cltp 6391 . . . . . . . . . 10 class <P
3027, 28, 29wbr 3764 . . . . . . . . 9 wff (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)
3125, 30wa 97 . . . . . . . 8 wff ((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3231, 20wex 1381 . . . . . . 7 wff 𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3332, 18wex 1381 . . . . . 6 wff 𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3433, 12wex 1381 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
3534, 10wex 1381 . . . 4 wff 𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
369, 35wa 97 . . 3 wff ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
3736, 2, 6copab 3817 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
381, 37wceq 1243 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 set class
This definition is referenced by:  ltrelsr  6821  ltsrprg  6830
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