Theorem chj12

Description: A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	chj12	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		chjcom	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}A$$
2	1	3adant3	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}B=B{\vee }_{\mathscr{H}}A$$
3	2	oveq1d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(B{\vee }_{\mathscr{H}}A\right){\vee }_{\mathscr{H}}C$$
4		chjass	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$
5		chjass	$$⊢\left(B\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{\vee }_{\mathscr{H}}A\right){\vee }_{\mathscr{H}}C=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$
6	5	3com12	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(B{\vee }_{\mathscr{H}}A\right){\vee }_{\mathscr{H}}C=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$
7	3 4 6	3eqtr3d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=B{\vee }_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}C\right)$$