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Definition df-leop 28839
Description: Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-leop op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
Distinct variable group:   𝑢,𝑡,𝑥

Detailed syntax breakdown of Definition df-leop
StepHypRef Expression
1 cleo 27943 . 2 class op
2 vu . . . . . . 7 setvar 𝑢
32cv 1522 . . . . . 6 class 𝑢
4 vt . . . . . . 7 setvar 𝑡
54cv 1522 . . . . . 6 class 𝑡
6 chod 27925 . . . . . 6 class op
73, 5, 6co 6690 . . . . 5 class (𝑢op 𝑡)
8 cho 27935 . . . . 5 class HrmOp
97, 8wcel 2030 . . . 4 wff (𝑢op 𝑡) ∈ HrmOp
10 cc0 9974 . . . . . 6 class 0
11 vx . . . . . . . . 9 setvar 𝑥
1211cv 1522 . . . . . . . 8 class 𝑥
1312, 7cfv 5926 . . . . . . 7 class ((𝑢op 𝑡)‘𝑥)
14 csp 27907 . . . . . . 7 class ·ih
1513, 12, 14co 6690 . . . . . 6 class (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
16 cle 10113 . . . . . 6 class
1710, 15, 16wbr 4685 . . . . 5 wff 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
18 chil 27904 . . . . 5 class
1917, 11, 18wral 2941 . . . 4 wff 𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
209, 19wa 383 . . 3 wff ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))
2120, 4, 2copab 4745 . 2 class {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
221, 21wceq 1523 1 wff op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
Colors of variables: wff setvar class
This definition is referenced by:  leopg  29109
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