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

Theorem leopg 32151
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 7439 . . . 4 (𝑡 = 𝑇 → (𝑢op 𝑡) = (𝑢op 𝑇))
21eleq1d 2824 . . 3 (𝑡 = 𝑇 → ((𝑢op 𝑡) ∈ HrmOp ↔ (𝑢op 𝑇) ∈ HrmOp))
31fveq1d 6909 . . . . . 6 (𝑡 = 𝑇 → ((𝑢op 𝑡)‘𝑥) = ((𝑢op 𝑇)‘𝑥))
43oveq1d 7446 . . . . 5 (𝑡 = 𝑇 → (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢op 𝑇)‘𝑥) ·ih 𝑥))
54breq2d 5160 . . . 4 (𝑡 = 𝑇 → (0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
65ralbidv 3176 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
72, 6anbi12d 632 . 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 7438 . . . 4 (𝑢 = 𝑈 → (𝑢op 𝑇) = (𝑈op 𝑇))
98eleq1d 2824 . . 3 (𝑢 = 𝑈 → ((𝑢op 𝑇) ∈ HrmOp ↔ (𝑈op 𝑇) ∈ HrmOp))
108fveq1d 6909 . . . . . 6 (𝑢 = 𝑈 → ((𝑢op 𝑇)‘𝑥) = ((𝑈op 𝑇)‘𝑥))
1110oveq1d 7446 . . . . 5 (𝑢 = 𝑈 → (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))
1211breq2d 5160 . . . 4 (𝑢 = 𝑈 → (0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
1312ralbidv 3176 . . 3 (𝑢 = 𝑈 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
149, 13anbi12d 632 . 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 31881 . 2 op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
167, 14, 15brabg 5549 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  0cc0 11153  cle 11294  chba 30948   ·ih csp 30951  op chod 30969  HrmOpcho 30979  op cleo 30987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-leop 31881
This theorem is referenced by:  leop  32152  leoprf2  32156
  Copyright terms: Public domain W3C validator