Definition df-ltr 11115

Description: Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 11177, 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⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((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

Step	Hyp	Ref	Expression
1		cltr 10927	. 2 class <R
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1569	. . . . . 6 class 𝑥
4		cnr 10921	. . . . . 6 class R
5	3, 4	wcel 2145	. . . . 5 wff 𝑥 ∈ R
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1569	. . . . . 6 class 𝑦
8	7, 4	wcel 2145	. . . . 5 wff 𝑦 ∈ R
9	5, 8	wa 401	. . . 4 wff (𝑥 ∈ R ∧ 𝑦 ∈ R)
10		vz	. . . . . . . . . . . . . 14 setvar 𝑧
11	10	cv 1569	. . . . . . . . . . . . 13 class 𝑧
12		vw	. . . . . . . . . . . . . 14 setvar 𝑤
13	12	cv 1569	. . . . . . . . . . . . 13 class 𝑤
14	11, 13	cop 4589	. . . . . . . . . . . 12 class ⟨𝑧, 𝑤⟩
15		cer 10920	. . . . . . . . . . . 12 class ~R
16	14, 15	cec 8693	. . . . . . . . . . 11 class [⟨𝑧, 𝑤⟩] ~R
17	3, 16	wceq 1570	. . . . . . . . . 10 wff 𝑥 = [⟨𝑧, 𝑤⟩] ~R
18		vv	. . . . . . . . . . . . . 14 setvar 𝑣
19	18	cv 1569	. . . . . . . . . . . . 13 class 𝑣
20		vu	. . . . . . . . . . . . . 14 setvar 𝑢
21	20	cv 1569	. . . . . . . . . . . . 13 class 𝑢
22	19, 21	cop 4589	. . . . . . . . . . . 12 class ⟨𝑣, 𝑢⟩
23	22, 15	cec 8693	. . . . . . . . . . 11 class [⟨𝑣, 𝑢⟩] ~R
24	7, 23	wceq 1570	. . . . . . . . . 10 wff 𝑦 = [⟨𝑣, 𝑢⟩] ~R
25	17, 24	wa 401	. . . . . . . . 9 wff (𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R )
26		cpp 10917	. . . . . . . . . . 11 class +P
27	11, 21, 26	co 7408	. . . . . . . . . 10 class (𝑧 +P 𝑢)
28	13, 19, 26	co 7408	. . . . . . . . . 10 class (𝑤 +P 𝑣)
29		cltp 10919	. . . . . . . . . 10 class <P
30	27, 28, 29	wbr 5102	. . . . . . . . 9 wff (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)
31	25, 30	wa 401	. . . . . . . 8 wff ((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
32	31, 20	wex 1812	. . . . . . 7 wff ∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
33	32, 18	wex 1812	. . . . . 6 wff ∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
34	33, 12	wex 1812	. . . . 5 wff ∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
35	34, 10	wex 1812	. . . 4 wff ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
36	9, 35	wa 401	. . 3 wff ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))
37	36, 2, 6	copab 5166	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
38	1, 37	wceq 1570	1 wff <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}

This definition is used by:  ltrelsr  11124  ltsrpr  11133