Theorem thlleval

Description: Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015)

	Ref	Expression
Hypotheses	thlval.k	⊢ 𝐾 = ( toHL ‘ 𝑊 )
	thlbas.c	⊢ 𝐶 = ( ClSubSp ‘ 𝑊 )
	thlleval.l	⊢ ≤ = ( le ‘ 𝐾 )
Assertion	thlleval	⊢ ( ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶 ) → ( 𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		thlval.k	⊢ 𝐾 = ( toHL ‘ 𝑊 )
2		thlbas.c	⊢ 𝐶 = ( ClSubSp ‘ 𝑊 )
3		thlleval.l	⊢ ≤ = ( le ‘ 𝐾 )
4	2	fvexi	⊢ 𝐶 ∈ V
5		eqid	⊢ ( toInc ‘ 𝐶 ) = ( toInc ‘ 𝐶 )
6		eqid	⊢ ( le ‘ ( toInc ‘ 𝐶 ) ) = ( le ‘ ( toInc ‘ 𝐶 ) )
7	1 2 5 6	thlle	⊢ ( le ‘ ( toInc ‘ 𝐶 ) ) = ( le ‘ 𝐾 )
8	3 7	eqtr4i	⊢ ≤ = ( le ‘ ( toInc ‘ 𝐶 ) )
9	5 8	ipole	⊢ ( ( 𝐶 ∈ V ∧ 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶 ) → ( 𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇 ) )
10	4 9	mp3an1	⊢ ( ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶 ) → ( 𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇 ) )