Theorem leoprf

Description: The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	leoprf	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		0le0	⊢ 0 ≤ 0
2		hmopf	⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ )
3		hodid	⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝑇 −op 𝑇 ) = 0hop )
4	2 3	syl	⊢ ( 𝑇 ∈ HrmOp → ( 𝑇 −op 𝑇 ) = 0hop )
5	4	adantr	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( 𝑇 −op 𝑇 ) = 0hop )
6	5	fveq1d	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) = ( 0hop ‘ 𝑥 ) )
7		ho0val	⊢ ( 𝑥 ∈ ℋ → ( 0hop ‘ 𝑥 ) = 0ℎ )
8	7	adantl	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( 0hop ‘ 𝑥 ) = 0ℎ )
9	6 8	eqtrd	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) = 0ℎ )
10	9	oveq1d	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) = ( 0ℎ ·ih 𝑥 ) )
11		hi01	⊢ ( 𝑥 ∈ ℋ → ( 0ℎ ·ih 𝑥 ) = 0 )
12	11	adantl	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( 0ℎ ·ih 𝑥 ) = 0 )
13	10 12	eqtr2d	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 0 = ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
14	1 13	breqtrid	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → 0 ≤ ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
15	14	ralrimiva	⊢ ( 𝑇 ∈ HrmOp → ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) )
16		leop	⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ) → ( 𝑇 ≤op 𝑇 ↔ ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) ) )
17	16	anidms	⊢ ( 𝑇 ∈ HrmOp → ( 𝑇 ≤op 𝑇 ↔ ∀ 𝑥 ∈ ℋ 0 ≤ ( ( ( 𝑇 −op 𝑇 ) ‘ 𝑥 ) ·ih 𝑥 ) ) )
18	15 17	mpbird	⊢ ( 𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇 )