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Theorem leopg 29311
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 6822 . . . 4 (𝑡 = 𝑇 → (𝑢op 𝑡) = (𝑢op 𝑇))
21eleq1d 2824 . . 3 (𝑡 = 𝑇 → ((𝑢op 𝑡) ∈ HrmOp ↔ (𝑢op 𝑇) ∈ HrmOp))
31fveq1d 6355 . . . . . 6 (𝑡 = 𝑇 → ((𝑢op 𝑡)‘𝑥) = ((𝑢op 𝑇)‘𝑥))
43oveq1d 6829 . . . . 5 (𝑡 = 𝑇 → (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢op 𝑇)‘𝑥) ·ih 𝑥))
54breq2d 4816 . . . 4 (𝑡 = 𝑇 → (0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
65ralbidv 3124 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
72, 6anbi12d 749 . 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 6821 . . . 4 (𝑢 = 𝑈 → (𝑢op 𝑇) = (𝑈op 𝑇))
98eleq1d 2824 . . 3 (𝑢 = 𝑈 → ((𝑢op 𝑇) ∈ HrmOp ↔ (𝑈op 𝑇) ∈ HrmOp))
108fveq1d 6355 . . . . . 6 (𝑢 = 𝑈 → ((𝑢op 𝑇)‘𝑥) = ((𝑈op 𝑇)‘𝑥))
1110oveq1d 6829 . . . . 5 (𝑢 = 𝑈 → (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))
1211breq2d 4816 . . . 4 (𝑢 = 𝑈 → (0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
1312ralbidv 3124 . . 3 (𝑢 = 𝑈 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
149, 13anbi12d 749 . 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 29041 . 2 op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
167, 14, 15brabg 5144 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  cfv 6049  (class class class)co 6814  0cc0 10148  cle 10287  chil 28106   ·ih csp 28109  op chod 28127  HrmOpcho 28137  op cleo 28145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-iota 6012  df-fv 6057  df-ov 6817  df-leop 29041
This theorem is referenced by:  leop  29312  leoprf2  29316
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