Theorem mdsym

Description: M-symmetry of the Hilbert lattice. Lemma 5 of Maeda p. 168. (Contributed by NM, 6-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	mdsym	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}B\leftrightarrow B{𝑀}_{\mathscr{H}}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(A{𝑀}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}B\right)$$
2		breq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(B{𝑀}_{\mathscr{H}}A\leftrightarrow B{𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)$$
3	1 2	bibi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\to \left(\left(A{𝑀}_{\mathscr{H}}B\leftrightarrow B{𝑀}_{\mathscr{H}}A\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}B\leftrightarrow B{𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)\right)$$
4		breq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\right)$$
5		breq1	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(B{𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\leftrightarrow if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)$$
6	4 5	bibi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}B\leftrightarrow B{𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\leftrightarrow if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\right)\right)$$
7		ifchhv	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
8		ifchhv	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
9	7 8	mdsymi	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right)\leftrightarrow if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,\mathscr{H}\right){𝑀}_{\mathscr{H}}if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,\mathscr{H}\right)$$
10	3 6 9	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝑀}_{\mathscr{H}}B\leftrightarrow B{𝑀}_{\mathscr{H}}A\right)$$