Theorem leopg 30385

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	⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 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.

Step	Hyp	Ref	Expression
1		oveq2 7263	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑢 −op 𝑡) = (𝑢 −op 𝑇))
2	1	eleq1d 2823	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡) ∈ HrmOp ↔ (𝑢 −op 𝑇) ∈ HrmOp))
3	1	fveq1d 6758	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡)‘𝑥) = ((𝑢 −op 𝑇)‘𝑥))
4	3	oveq1d 7270	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))
5	4	breq2d 5082	. . . 4 ⊢ (𝑡 = 𝑇 → (0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)))
6	5	ralbidv 3120	. . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)))
7	2, 6	anbi12d 630	. 2 ⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥)) ↔ ((𝑢 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))))
8		oveq1 7262	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 −op 𝑇) = (𝑈 −op 𝑇))
9	8	eleq1d 2823	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇) ∈ HrmOp ↔ (𝑈 −op 𝑇) ∈ HrmOp))
10	8	fveq1d 6758	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇)‘𝑥) = ((𝑈 −op 𝑇)‘𝑥))
11	10	oveq1d 7270	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))
12	11	breq2d 5082	. . . 4 ⊢ (𝑢 = 𝑈 → (0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
13	12	ralbidv 3120	. . 3 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
14	9, 13	anbi12d 630	. 2 ⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)) ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))))
15		df-leop 30115	. 2 ⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
16	7, 14, 15	brabg 5445	1 ⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  0cc0 10802   ≤ cle 10941   ℋchba 29182   ·_ih csp 29185   −_op chod 29203  HrmOpcho 29213   ≤_op cleo 29221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258  df-leop 30115

This theorem is referenced by:  leop  30386  leoprf2  30390