HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  df-leop Structured version   Visualization version   GIF version

Definition df-leop 29933
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 29039 . 2 class op
2 vu . . . . . . 7 setvar 𝑢
32cv 1542 . . . . . 6 class 𝑢
4 vt . . . . . . 7 setvar 𝑡
54cv 1542 . . . . . 6 class 𝑡
6 chod 29021 . . . . . 6 class op
73, 5, 6co 7213 . . . . 5 class (𝑢op 𝑡)
8 cho 29031 . . . . 5 class HrmOp
97, 8wcel 2110 . . . 4 wff (𝑢op 𝑡) ∈ HrmOp
10 cc0 10729 . . . . . 6 class 0
11 vx . . . . . . . . 9 setvar 𝑥
1211cv 1542 . . . . . . . 8 class 𝑥
1312, 7cfv 6380 . . . . . . 7 class ((𝑢op 𝑡)‘𝑥)
14 csp 29003 . . . . . . 7 class ·ih
1513, 12, 14co 7213 . . . . . 6 class (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
16 cle 10868 . . . . . 6 class
1710, 15, 16wbr 5053 . . . . 5 wff 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
18 chba 29000 . . . . 5 class
1917, 11, 18wral 3061 . . . 4 wff 𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)
209, 19wa 399 . . 3 wff ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))
2120, 4, 2copab 5115 . 2 class {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
221, 21wceq 1543 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  30203
  Copyright terms: Public domain W3C validator