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

Theorem leopg 29909
Description: Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
leopg ((𝑇𝐴𝑈𝐵) → (𝑇op 𝑈 ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑇   𝑥,𝑈

Proof of Theorem leopg
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7147 . . . 4 (𝑡 = 𝑇 → (𝑢op 𝑡) = (𝑢op 𝑇))
21eleq1d 2877 . . 3 (𝑡 = 𝑇 → ((𝑢op 𝑡) ∈ HrmOp ↔ (𝑢op 𝑇) ∈ HrmOp))
31fveq1d 6651 . . . . . 6 (𝑡 = 𝑇 → ((𝑢op 𝑡)‘𝑥) = ((𝑢op 𝑇)‘𝑥))
43oveq1d 7154 . . . . 5 (𝑡 = 𝑇 → (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢op 𝑇)‘𝑥) ·ih 𝑥))
54breq2d 5045 . . . 4 (𝑡 = 𝑇 → (0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
65ralbidv 3165 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
72, 6anbi12d 633 . 2 (𝑡 = 𝑇 → (((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥)) ↔ ((𝑢op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥))))
8 oveq1 7146 . . . 4 (𝑢 = 𝑈 → (𝑢op 𝑇) = (𝑈op 𝑇))
98eleq1d 2877 . . 3 (𝑢 = 𝑈 → ((𝑢op 𝑇) ∈ HrmOp ↔ (𝑈op 𝑇) ∈ HrmOp))
108fveq1d 6651 . . . . . 6 (𝑢 = 𝑈 → ((𝑢op 𝑇)‘𝑥) = ((𝑈op 𝑇)‘𝑥))
1110oveq1d 7154 . . . . 5 (𝑢 = 𝑈 → (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))
1211breq2d 5045 . . . 4 (𝑢 = 𝑈 → (0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
1312ralbidv 3165 . . 3 (𝑢 = 𝑈 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
149, 13anbi12d 633 . 2 (𝑢 = 𝑈 → (((𝑢op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)) ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))
15 df-leop 29639 . 2 op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
167, 14, 15brabg 5394 1 ((𝑇𝐴𝑈𝐵) → (𝑇op 𝑈 ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109   class class class wbr 5033  cfv 6328  (class class class)co 7139  0cc0 10530  cle 10669  chba 28706   ·ih csp 28709  op chod 28727  HrmOpcho 28737  op cleo 28745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-iota 6287  df-fv 6336  df-ov 7142  df-leop 29639
This theorem is referenced by:  leop  29910  leoprf2  29914
  Copyright terms: Public domain W3C validator