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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   ℋchba 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
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