Theorem dmdsym

Description: Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	dmdsym	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow B{𝑀}_{\mathscr{H}}^{*}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		choccl	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
2		choccl	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}\to \perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
3		mdsym	$$⊢\left(\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}\wedge \perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)\leftrightarrow \perp \left(B\right){𝑀}_{\mathscr{H}}\perp \left(A\right)\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(\perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)\leftrightarrow \perp \left(B\right){𝑀}_{\mathscr{H}}\perp \left(A\right)\right)$$
5		dmdmd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow \perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)\right)$$
6		dmdmd	$$⊢\left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{𝑀}_{\mathscr{H}}^{*}A\leftrightarrow \perp \left(B\right){𝑀}_{\mathscr{H}}\perp \left(A\right)\right)$$
7	6	ancoms	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{𝑀}_{\mathscr{H}}^{*}A\leftrightarrow \perp \left(B\right){𝑀}_{\mathscr{H}}\perp \left(A\right)\right)$$
8	4 5 7	3bitr4d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow B{𝑀}_{\mathscr{H}}^{*}A\right)$$