Theorem chjjdiri

Description: Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chj12.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chj12.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
	chj12.3	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chjjdiri	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		chj12.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chj12.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chj12.3	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
4	3	chjidmi	$$⊢C{\vee }_{\mathscr{H}}C=C$$
5	4	oveq2i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}C\right)=\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C$$
6	1 2 3 3	chj4i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}C\right)=\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$
7	5 6	eqtr3i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$