Theorem chjass

Description: Associative law for Hilbert lattice join. From definition of lattice in Kalmbach p. 14. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		oveq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to A{\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B$$
2	1	oveq1d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C$$
3		oveq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$
4	2 3	eqeq12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)\right)$$
5		oveq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)$$
6	5	oveq1d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}C$$
7		oveq1	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to B{\vee }_{\mathscr{H}}C=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C$$
8	7	oveq2d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C\right)$$
9	6 8	eqeq12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}C=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C\right)\right)$$
10		oveq2	$$⊢C=if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}C=\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)$$
11		oveq2	$$⊢C=if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\to if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)$$
12	11	oveq2d	$$⊢C=if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\to if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C\right)=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\right)$$
13	10 12	eqeq12d	$$⊢C=if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}C=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}C\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\right)\right)$$
14		ifchhv	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
15		ifchhv	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
16		ifchhv	$$⊢if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
17	14 15 16	chjassi	$$⊢\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){\vee }_{\mathscr{H}}\left(if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){\vee }_{\mathscr{H}}if\left(C\in {\mathbf{C}}_{\mathscr{H}},C,\mathscr{H}\right)\right)$$
18	4 9 13 17	dedth3h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{\vee }_{\mathscr{H}}B\right){\vee }_{\mathscr{H}}C=A{\vee }_{\mathscr{H}}\left(B{\vee }_{\mathscr{H}}C\right)$$