Theorem cmcm3

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	cmcm3	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq1	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\right)$$
2		fveq2	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \perp \left(A\right)=\perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right)$$
3	2	breq1d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\perp \left(A\right){𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}B\right)$$
4	1 3	bibi12d	$$⊢A=if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\to \left(\left(A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}B\right)\right)$$
5		breq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\leftrightarrow if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
6		breq2	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)$$
7	5 6	bibi12d	$$⊢B=if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\to \left(\left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}B\right)\leftrightarrow \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\right)\right)$$
8		h0elch	$$⊢0_{\mathscr{H}}\in {\mathbf{C}}_{\mathscr{H}}$$
9	8	elimel	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
10	8	elimel	$$⊢if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
11	9 10	cmcm3i	$$⊢if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)\leftrightarrow \perp \left(if\left(A\in {\mathbf{C}}_{\mathscr{H}},A,0_{\mathscr{H}}\right)\right){𝐶}_{\mathscr{H}}if\left(B\in {\mathbf{C}}_{\mathscr{H}},B,0_{\mathscr{H}}\right)$$
12	4 7 11	dedth2h	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A{𝐶}_{\mathscr{H}}B\leftrightarrow \perp \left(A\right){𝐶}_{\mathscr{H}}B\right)$$