Theorem atord

Description: An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	atord	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\wedge A{𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq A\vee B\subseteq \perp \left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)$$
2	1	anbi2d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(B\in \mathrm{HAtoms}\wedge A{𝐶}_{\mathscr{H}}B\right)\leftrightarrow \left(B\in \mathrm{HAtoms}\wedge if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)\right)$$
3		sseq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(B\subseteq A\leftrightarrow B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
4		fveq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \perp \left(A\right)=\perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
5	4	sseq2d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(B\subseteq \perp \left(A\right)\leftrightarrow B\subseteq \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)\right)$$
6	3 5	orbi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(B\subseteq A\vee B\subseteq \perp \left(A\right)\right)\leftrightarrow \left(B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\vee B\subseteq \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)\right)\right)$$
7	2 6	imbi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(\left(B\in \mathrm{HAtoms}\wedge A{𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq A\vee B\subseteq \perp \left(A\right)\right)\right)\leftrightarrow \left(\left(B\in \mathrm{HAtoms}\wedge if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\vee B\subseteq \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)\right)\right)\right)$$
8		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
9	8	elimel	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
10	9	atordi	$$⊢\left(B\in \mathrm{HAtoms}\wedge if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\vee B\subseteq \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)\right)$$
11	7 10	dedth	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(\left(B\in \mathrm{HAtoms}\wedge A{𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq A\vee B\subseteq \perp \left(A\right)\right)\right)$$
12	11	3impib	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\wedge A{𝐶}_{\mathscr{H}}B\right)\to \left(B\subseteq A\vee B\subseteq \perp \left(A\right)\right)$$