Theorem pjeq

Description: Equality with a projection. (Contributed by NM, 20-Jan-2007) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	pjeq	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=B\leftrightarrow \left(B\in H\wedge \exists x\in \perp \left(H\right)A=B{+}_{ℎ}x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjhth	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H{+}_{\mathscr{H}}\perp \left(H\right)=\mathscr{H}$$
2	1	eleq2d	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to \left(A\in \left(H{+}_{\mathscr{H}}\perp \left(H\right)\right)\leftrightarrow A\in \mathscr{H}\right)$$
3	2	biimpar	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to A\in \left(H{+}_{\mathscr{H}}\perp \left(H\right)\right)$$
4		pjpreeq	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \left(H{+}_{\mathscr{H}}\perp \left(H\right)\right)\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=B\leftrightarrow \left(B\in H\wedge \exists x\in \perp \left(H\right)A=B{+}_{ℎ}x\right)\right)$$
5	3 4	syldan	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in \mathscr{H}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)=B\leftrightarrow \left(B\in H\wedge \exists x\in \perp \left(H\right)A=B{+}_{ℎ}x\right)\right)$$