Theorem dmdi4

Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	dmdi4	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\to \left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		dmdbr4	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\leftrightarrow \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$
2	1	biimpd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\to \forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$
3		oveq1	$$⊢x=C\to x{\vee }_{\mathscr{H}}B=C{\vee }_{\mathscr{H}}B$$
4	3	ineq1d	$$⊢x=C\to \left(x{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)=\left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)$$
5	3	ineq1d	$$⊢x=C\to \left(x{\vee }_{\mathscr{H}}B\right)\cap A=\left(C{\vee }_{\mathscr{H}}B\right)\cap A$$
6	5	oveq1d	$$⊢x=C\to \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B=\left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B$$
7	4 6	sseq12d	$$⊢x=C\to \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\leftrightarrow \left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$
8	7	rspcv	$$⊢C\in {\mathbf{C}}_{\mathscr{H}}\to \left(\forall x\in {\mathbf{C}}_{\mathscr{H}}\left(x{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(x{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\to \left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$
9	2 8	sylan9	$$⊢\left(\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\to \left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$
10	9	3impa	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\wedge C\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}^{*}B\to \left(C{\vee }_{\mathscr{H}}B\right)\cap \left(A{\vee }_{\mathscr{H}}B\right)\subseteq \left(\left(C{\vee }_{\mathscr{H}}B\right)\cap A\right){\vee }_{\mathscr{H}}B\right)$$