Theorem chj12i

Description: A rearrangement of Hilbert lattice join. (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	chj12i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$

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	1 2	chjcomi	$$⊢A{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}A$$
5	4	oveq1i	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(B{\vee }_{\mathscr{H}}A\right){\vee }_{\mathscr{H}}C$$
6	1 2 3	chjassi	$$⊢\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$
7	2 1 3	chjassi	$$⊢\left(B{\vee }_{\mathscr{H}}A\right){\vee }_{\mathscr{H}}C=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$
8	5 6 7	3eqtr3i	$$⊢A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$