Theorem chlejb1

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlejb1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sseq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\right)$$
2		oveq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B$$
3	2	eqeq1d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A{\vee }_{\mathscr{H}}B=B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=B\right)$$
4	1 3	bibi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=B\right)\right)$$
5		sseq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
6		oveq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
7		id	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
8	6 7	eqeq12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
9	5 8	bibi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}B=B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)\right)$$
10		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
11	10	elimel	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
12	10	elimel	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
13	11 12	chlejb1i	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\subseteq if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
14	4 9 13	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow A{\vee }_{\mathscr{H}}B=B\right)$$