Theorem pjssge0i

Description: Theorem 4.5(iv)->(v) 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	pjssge0i	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\to 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right){\cdot }_{\mathrm{ih}}A\right)$$

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		id	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)$$
9	5 8	oveq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right){\cdot }_{\mathrm{ih}}A=\left({\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)$$
10	9	breq2d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left(0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right){\cdot }_{\mathrm{ih}}A\leftrightarrow 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
11	7 10	imbi12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left(\left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\to 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right){\cdot }_{\mathrm{ih}}A\right)\leftrightarrow \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\to 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)$$
12		ifhvhv0	$$⊢if\left(A\in \mathscr{H},A,0_{ℎ}\right)\in \mathscr{H}$$
13	2 12 1	pjssge0ii	$$⊢{\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\to 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)$$
14	11 13	dedth	$$⊢A\in \mathscr{H}\to \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(G\cap \perp \left(H\right)\right)\left(A\right)\to 0\le \left({\mathrm{proj}}_{ℎ}\left(G\right)\left(A\right){-}_{ℎ}{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right){\cdot }_{\mathrm{ih}}A\right)$$