Theorem osumcori

Description: Corollary of osumi . (Contributed by NM, 5-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	osum.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	osum.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	osumcori	$$⊢\left(A\cap B\right){+}_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		osum.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		osum.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		inss2	$$⊢A\cap B\subseteq B$$
4	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	2 4	chub2i	$$⊢B\subseteq \perp \left(A\right){\vee }_{\mathscr{H}}B$$
6	3 5	sstri	$$⊢A\cap B\subseteq \perp \left(A\right){\vee }_{\mathscr{H}}B$$
7	1 2	chdmm3i	$$⊢\perp \left(A\cap \perp \left(B\right)\right)=\perp \left(A\right){\vee }_{\mathscr{H}}B$$
8	6 7	sseqtrri	$$⊢A\cap B\subseteq \perp \left(A\cap \perp \left(B\right)\right)$$
9	1 2	chincli	$$⊢A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$
10	2	choccli	$$⊢\perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
11	1 10	chincli	$$⊢A\cap \perp \left(B\right)\in {\mathbf{C}}_{\mathscr{H}}$$
12	9 11	osumi	$$⊢A\cap B\subseteq \perp \left(A\cap \perp \left(B\right)\right)\to \left(A\cap B\right){+}_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$
13	8 12	ax-mp	$$⊢\left(A\cap B\right){+}_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)=\left(A\cap B\right){\vee }_{\mathscr{H}}\left(A\cap \perp \left(B\right)\right)$$