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Theorem leopg 32141
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 2826 . . 3 (𝑡 = 𝑇 → ((𝑢op 𝑡) ∈ HrmOp ↔ (𝑢op 𝑇) ∈ HrmOp))
31fveq1d 6908 . . . . . 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 5155 . . . 4 (𝑡 = 𝑇 → (0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥)))
65ralbidv 3178 . . 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 2826 . . 3 (𝑢 = 𝑈 → ((𝑢op 𝑇) ∈ HrmOp ↔ (𝑈op 𝑇) ∈ HrmOp))
108fveq1d 6908 . . . . . 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 5155 . . . 4 (𝑢 = 𝑈 → (0 ≤ (((𝑢op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
1312ralbidv 3178 . . 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 31871 . 2 op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}
167, 14, 15brabg 5544 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 1540  wcel 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  0cc0 11155  cle 11296  chba 30938   ·ih csp 30941  op chod 30959  HrmOpcho 30969  op cleo 30977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-leop 31871
This theorem is referenced by:  leop  32142  leoprf2  32146
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