Theorem cmmdi

Description: Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)

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

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 2	cmcm4i	$$⊢A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)$$
4	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6	4 5	osumcor2i	$$⊢\perp \left(A\right){𝐶}_{\mathscr{H}}\perp \left(B\right)\to \perp \left(A\right){+}_{\mathscr{H}}\perp \left(B\right)=\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)$$
7	3 6	sylbi	$$⊢A{𝐶}_{\mathscr{H}}B\to \perp \left(A\right){+}_{\mathscr{H}}\perp \left(B\right)=\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)$$
8	4 5	sumdmdii	$$⊢\perp \left(A\right){+}_{\mathscr{H}}\perp \left(B\right)=\perp \left(A\right){\vee }_{\mathscr{H}}\perp \left(B\right)\to \perp \left(A\right){𝑀}_{\mathscr{H}}^{*}\perp \left(B\right)$$
9	7 8	syl	$$⊢A{𝐶}_{\mathscr{H}}B\to \perp \left(A\right){𝑀}_{\mathscr{H}}^{*}\perp \left(B\right)$$
10		mddmd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝑀}_{\mathscr{H}}^{*}\perp \left(B\right)\right)$$
11	1 2 10	mp2an	$$⊢A{𝑀}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝑀}_{\mathscr{H}}^{*}\perp \left(B\right)$$
12	9 11	sylibr	$$⊢A{𝐶}_{\mathscr{H}}B\to A{𝑀}_{\mathscr{H}}B$$