Theorem stji1i

Description: Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	stle.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	stle.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	stji1i	$$⊢S\in \mathrm{States}\to S\left(\perp \left(A\right){\vee }_{\mathscr{H}}\left(A\cap B\right)\right)=S\left(\perp \left(A\right)\right)+S\left(A\cap B\right)$$

Proof

Step	Hyp	Ref	Expression
1		stle.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		stle.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	1 2	chincli	$$⊢A\cap B\in {\mathbf{C}}_{\mathscr{H}}$$
5	3 4	pm3.2i	$$⊢\left(\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}\wedge A\cap B\in {\mathbf{C}}_{\mathscr{H}}\right)$$
6		inss1	$$⊢A\cap B\subseteq A$$
7	4 1	chsscon3i	$$⊢A\cap B\subseteq A\leftrightarrow \perp \left(A\right)\subseteq \perp \left(A\cap B\right)$$
8	6 7	mpbi	$$⊢\perp \left(A\right)\subseteq \perp \left(A\cap B\right)$$
9		stj	$$⊢S\in \mathrm{States}\to \left(\left(\left(\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}\wedge A\cap B\in {\mathbf{C}}_{\mathscr{H}}\right)\wedge \perp \left(A\right)\subseteq \perp \left(A\cap B\right)\right)\to S\left(\perp \left(A\right){\vee }_{\mathscr{H}}\left(A\cap B\right)\right)=S\left(\perp \left(A\right)\right)+S\left(A\cap B\right)\right)$$
10	5 8 9	mp2ani	$$⊢S\in \mathrm{States}\to S\left(\perp \left(A\right){\vee }_{\mathscr{H}}\left(A\cap B\right)\right)=S\left(\perp \left(A\right)\right)+S\left(A\cap B\right)$$