Metamath Proof Explorer


Definition df-ltr

Description: Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c , 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.)

Ref Expression
Assertion df-ltr <R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥R𝑦R ) ∧ ∃ 𝑧𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cltr <R
1 vx 𝑥
2 vy 𝑦
3 1 cv 𝑥
4 cnr R
5 3 4 wcel 𝑥R
6 2 cv 𝑦
7 6 4 wcel 𝑦R
8 5 7 wa ( 𝑥R𝑦R )
9 vz 𝑧
10 vw 𝑤
11 vv 𝑣
12 vu 𝑢
13 9 cv 𝑧
14 10 cv 𝑤
15 13 14 cop 𝑧 , 𝑤
16 cer ~R
17 15 16 cec [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R
18 3 17 wceq 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R
19 11 cv 𝑣
20 12 cv 𝑢
21 19 20 cop 𝑣 , 𝑢
22 21 16 cec [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R
23 6 22 wceq 𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R
24 18 23 wa ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R )
25 cpp +P
26 13 20 25 co ( 𝑧 +P 𝑢 )
27 cltp <P
28 14 19 25 co ( 𝑤 +P 𝑣 )
29 26 28 27 wbr ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 )
30 24 29 wa ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) )
31 30 12 wex 𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) )
32 31 11 wex 𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) )
33 32 10 wex 𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) )
34 33 9 wex 𝑧𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) )
35 8 34 wa ( ( 𝑥R𝑦R ) ∧ ∃ 𝑧𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) )
36 35 1 2 copab { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥R𝑦R ) ∧ ∃ 𝑧𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) ) }
37 0 36 wceq <R = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥R𝑦R ) ∧ ∃ 𝑧𝑤𝑣𝑢 ( ( 𝑥 = [ ⟨ 𝑧 , 𝑤 ⟩ ] ~R𝑦 = [ ⟨ 𝑣 , 𝑢 ⟩ ] ~R ) ∧ ( 𝑧 +P 𝑢 ) <P ( 𝑤 +P 𝑣 ) ) ) }