Theorem leopg 29883

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 7150	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑢 −op 𝑡) = (𝑢 −op 𝑇))
2	1	eleq1d 2897	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡) ∈ HrmOp ↔ (𝑢 −op 𝑇) ∈ HrmOp))
3	1	fveq1d 6658	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑢 −op 𝑡)‘𝑥) = ((𝑢 −op 𝑇)‘𝑥))
4	3	oveq1d 7157	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) = (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥))
5	4	breq2d 5064	. . . 4 ⊢ (𝑡 = 𝑇 → (0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥)))
6	5	ralbidv 3197	. . 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 7149	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 −op 𝑇) = (𝑈 −op 𝑇))
9	8	eleq1d 2897	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇) ∈ HrmOp ↔ (𝑈 −op 𝑇) ∈ HrmOp))
10	8	fveq1d 6658	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢 −op 𝑇)‘𝑥) = ((𝑈 −op 𝑇)‘𝑥))
11	10	oveq1d 7157	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) = (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))
12	11	breq2d 5064	. . . 4 ⊢ (𝑢 = 𝑈 → (0 ≤ (((𝑢 −op 𝑇)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
13	12	ralbidv 3197	. . 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 29613	. 2 ⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
16	7, 14, 15	brabg 5412	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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   class class class wbr 5052  ‘cfv 6341  (class class class)co 7142  0cc0 10523   ≤ cle 10662   ℋchba 28680   ·_ih csp 28683   −_op chod 28701  HrmOpcho 28711   ≤_op cleo 28719

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-iota 6300  df-fv 6349  df-ov 7145  df-leop 29613

This theorem is referenced by:  leop  29884  leoprf2  29888