Theorem thlleval

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

	Ref	Expression
Hypotheses	thlval.k	$$⊢K=\mathrm{toHL}\left(W\right)$$
	thlbas.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
	thlleval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	thlleval	$$⊢\left(S\in C\wedge T\in C\right)\to \left(S\underset{\dot{}}{\le }T\leftrightarrow S\subseteq T\right)$$

Proof

Step	Hyp	Ref	Expression
1		thlval.k	$$⊢K=\mathrm{toHL}\left(W\right)$$
2		thlbas.c	$$⊢C=\mathrm{ClSubSp}\left(W\right)$$
3		thlleval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
4	2	fvexi	$$⊢C\in \mathrm{V}$$
5		eqid	$$⊢\mathrm{toInc}\left(C\right)=\mathrm{toInc}\left(C\right)$$
6		eqid	$$⊢{\le }_{\mathrm{toInc}\left(C\right)}={\le }_{\mathrm{toInc}\left(C\right)}$$
7	1 2 5 6	thlle	$$⊢{\le }_{\mathrm{toInc}\left(C\right)}={\le }_K$$
8	3 7	eqtr4i	$$⊢\underset{\dot{}}{\le }={\le }_{\mathrm{toInc}\left(C\right)}$$
9	5 8	ipole	$$⊢\left(C\in \mathrm{V}\wedge S\in C\wedge T\in C\right)\to \left(S\underset{\dot{}}{\le }T\leftrightarrow S\subseteq T\right)$$
10	4 9	mp3an1	$$⊢\left(S\in C\wedge T\in C\right)\to \left(S\underset{\dot{}}{\le }T\leftrightarrow S\subseteq T\right)$$