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Definition df-ltr 6638
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⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨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 6287 . 2 class <R
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cnr 6281 . . . . . 6 class R
53, 4wcel 1390 . . . . 5 wff x R
6 vy . . . . . . 7 setvar y
76cv 1241 . . . . . 6 class y
87, 4wcel 1390 . . . . 5 wff y R
95, 8wa 97 . . . 4 wff (x R y R)
10 vz . . . . . . . . . . . . . 14 setvar z
1110cv 1241 . . . . . . . . . . . . 13 class z
12 vw . . . . . . . . . . . . . 14 setvar w
1312cv 1241 . . . . . . . . . . . . 13 class w
1411, 13cop 3370 . . . . . . . . . . . 12 class z, w
15 cer 6280 . . . . . . . . . . . 12 class ~R
1614, 15cec 6040 . . . . . . . . . . 11 class [⟨z, w⟩] ~R
173, 16wceq 1242 . . . . . . . . . 10 wff x = [⟨z, w⟩] ~R
18 vv . . . . . . . . . . . . . 14 setvar v
1918cv 1241 . . . . . . . . . . . . 13 class v
20 vu . . . . . . . . . . . . . 14 setvar u
2120cv 1241 . . . . . . . . . . . . 13 class u
2219, 21cop 3370 . . . . . . . . . . . 12 class v, u
2322, 15cec 6040 . . . . . . . . . . 11 class [⟨v, u⟩] ~R
247, 23wceq 1242 . . . . . . . . . 10 wff y = [⟨v, u⟩] ~R
2517, 24wa 97 . . . . . . . . 9 wff (x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R )
26 cpp 6277 . . . . . . . . . . 11 class +P
2711, 21, 26co 5455 . . . . . . . . . 10 class (z +P u)
2813, 19, 26co 5455 . . . . . . . . . 10 class (w +P v)
29 cltp 6279 . . . . . . . . . 10 class <P
3027, 28, 29wbr 3755 . . . . . . . . 9 wff (z +P u)<P (w +P v)
3125, 30wa 97 . . . . . . . 8 wff ((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v))
3231, 20wex 1378 . . . . . . 7 wff u((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v))
3332, 18wex 1378 . . . . . 6 wff vu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v))
3433, 12wex 1378 . . . . 5 wff wvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v))
3534, 10wex 1378 . . . 4 wff zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v))
369, 35wa 97 . . 3 wff ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))
3736, 2, 6copab 3808 . 2 class {⟨x, y⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))}
381, 37wceq 1242 1 wff <R = {⟨x, y⟩ ∣ ((x R y R) zwvu((x = [⟨z, w⟩] ~R y = [⟨v, u⟩] ~R ) (z +P u)<P (w +P v)))}
Colors of variables: wff set class
This definition is referenced by:  ltrelsr  6646  ltsrprg  6655
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