Theorem leopg 32146

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 7364	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑢 −op 𝑡) = (𝑢 −op 𝑇))
2	1	eleq1d 2819	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡) ∈ HrmOp ↔ (𝑢 −op 𝑇) ∈ HrmOp))
3	1	fveq1d 6834	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡)‘𝑥) = ((𝑢 −op 𝑇)‘𝑥))
4	3	oveq1d 7371	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))
5	4	breq2d 5108	. . . 4 ⊢ (𝑡 = 𝑇 → (0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)))
6	5	ralbidv 3157	. . 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 632	. 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 7363	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 −op 𝑇) = (𝑈 −op 𝑇))
9	8	eleq1d 2819	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇) ∈ HrmOp ↔ (𝑈 −op 𝑇) ∈ HrmOp))
10	8	fveq1d 6834	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇)‘𝑥) = ((𝑈 −op 𝑇)‘𝑥))
11	10	oveq1d 7371	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))
12	11	breq2d 5108	. . . 4 ⊢ (𝑢 = 𝑈 → (0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
13	12	ralbidv 3157	. . 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 632	. 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 31876	. 2 ⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
16	7, 14, 15	brabg 5485	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  0cc0 11024   ≤ cle 11165   ℋchba 30943   ·_ih csp 30946   −_op chod 30964  HrmOpcho 30974   ≤_op cleo 30982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-iota 6446  df-fv 6498  df-ov 7359  df-leop 31876

This theorem is referenced by:  leop  32147  leoprf2  32151