Theorem chj00i

Description: Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chj00i	$$⊢\left(A=0_{\mathscr{H}}\wedge B=0_{\mathscr{H}}\right)\leftrightarrow A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		oveq12	$$⊢\left(A=0_{\mathscr{H}}\wedge B=0_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}{\vee }_{\mathscr{H}}{0}_{\mathscr{H}}$$
4		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
5	4	chj0i	$$⊢0_{\mathscr{H}}{\vee }_{\mathscr{H}}{0}_{\mathscr{H}}=0_{\mathscr{H}}$$
6	3 5	eqtrdi	$$⊢\left(A=0_{\mathscr{H}}\wedge B=0_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}$$
7	1 2	chub1i	$$⊢A\subseteq A{\vee }_{\mathscr{H}}B$$
8		sseq2	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to \left(A\subseteq A{\vee }_{\mathscr{H}}B\leftrightarrow A\subseteq 0_{\mathscr{H}}\right)$$
9	7 8	mpbii	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to A\subseteq 0_{\mathscr{H}}$$
10	1	chle0i	$$⊢A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}$$
11	9 10	sylib	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to A=0_{\mathscr{H}}$$
12	2 1	chub2i	$$⊢B\subseteq A{\vee }_{\mathscr{H}}B$$
13		sseq2	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to \left(B\subseteq A{\vee }_{\mathscr{H}}B\leftrightarrow B\subseteq 0_{\mathscr{H}}\right)$$
14	12 13	mpbii	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to B\subseteq 0_{\mathscr{H}}$$
15	2	chle0i	$$⊢B\subseteq 0_{\mathscr{H}}\leftrightarrow B=0_{\mathscr{H}}$$
16	14 15	sylib	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to B=0_{\mathscr{H}}$$
17	11 16	jca	$$⊢A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}\to \left(A=0_{\mathscr{H}}\wedge B=0_{\mathscr{H}}\right)$$
18	6 17	impbii	$$⊢\left(A=0_{\mathscr{H}}\wedge B=0_{\mathscr{H}}\right)\leftrightarrow A{\vee }_{\mathscr{H}}B=0_{\mathscr{H}}$$