Theorem pjssmi

Description: Projection meet property. Remark in Kalmbach p. 66. Also Theorem 4.5(i)->(iv) of Beran p. 112. (Contributed by NM, 26-Sep-2001) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
	pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjssmi	$$⊢A\in \mathscr{H}\to \left(H\subseteq G\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjco.1	$$⊢G\in {\mathbf{C}}_{\mathscr{H}}$$
2		pjco.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3		fveq2	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
4		fveq2	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
5	3 4	oveq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
6		fveq2	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
7	5 6	eqeq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\leftrightarrow {\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)$$
8	7	imbi2d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left(\left(H\subseteq G\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\right)\leftrightarrow \left(H\subseteq G\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)\right)$$
9		ifhvhv0	$$⊢if\left(A\in \mathscr{H},A,0_{ℎ}\right)\in \mathscr{H}$$
10	2 9 1	pjssmii	$$⊢H\subseteq G\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
11	8 10	dedth	$$⊢A\in \mathscr{H}\to \left(H\subseteq G\to {\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\right)$$