Theorem cmdmdi

Description: Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sumdmdi.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	sumdmdi.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	cmdmdi	$$⊢A{𝐶}_{\mathscr{H}}B\to A{𝑀}_{\mathscr{H}}^{*}B$$

Proof

Step	Hyp	Ref	Expression
1		sumdmdi.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		sumdmdi.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	3 4	cmmdi	$$⊢\perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)\to \perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)$$
6	1 2	cmcm4i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)$$
7		dmdmd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow \perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)\right)$$
8	1 2 7	mp2an	$$⊢A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow \perp \left(A\right){𝑀}_{\mathscr{H}}\perp \left(B\right)$$
9	5 6 8	3imtr4i	$$⊢A{𝐶}_{\mathscr{H}}B\to A{𝑀}_{\mathscr{H}}^{*}B$$