Theorem chj4i

Description: Rearrangement of the join of 4 Hilbert lattice elements. (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	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}$$
	chj4.4	$$⊢D\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chj4i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)=\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)$$

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		chj4.4	$$⊢D\in {\mathbf{C}}_{\mathscr{H}}$$
5	2 3 4	chj12i	$$⊢B{\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)=C{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)$$
6	5	oveq2i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)\right)=A{\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)\right)$$
7	3 4	chjcli	$$⊢C{\vee }_{\mathscr{H}}D\in {\mathbf{C}}_{\mathscr{H}}$$
8	1 2 7	chjassi	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)=A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)\right)$$
9	2 4	chjcli	$$⊢B{\vee }_{\mathscr{H}}D\in {\mathbf{C}}_{\mathscr{H}}$$
10	1 3 9	chjassi	$$⊢\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)=A{\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)\right)$$
11	6 8 10	3eqtr4i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}\left(C{\vee }_{\mathscr{H}}D\right)=\left(A{\vee }_{\mathscr{H}}C\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}D\right)$$