# Metamath Proof Explorer

## Theorem chjo

Description: The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005) (New usage is discouraged.)

Ref Expression
Assertion chjo ${⊢}{A}\in {\mathbf{C}}_{ℋ}\to {A}{\vee }_{ℋ}\perp \left({A}\right)=ℋ$

### Proof

Step Hyp Ref Expression
1 id ${⊢}{A}=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\to {A}=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)$
2 fveq2 ${⊢}{A}=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\to \perp \left({A}\right)=\perp \left(if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\right)$
3 1 2 oveq12d ${⊢}{A}=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\to {A}{\vee }_{ℋ}\perp \left({A}\right)=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right){\vee }_{ℋ}\perp \left(if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\right)$
4 3 eqeq1d ${⊢}{A}=if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\to \left({A}{\vee }_{ℋ}\perp \left({A}\right)=ℋ↔if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right){\vee }_{ℋ}\perp \left(if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\right)=ℋ\right)$
5 ifchhv ${⊢}if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\in {\mathbf{C}}_{ℋ}$
6 5 chjoi ${⊢}if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right){\vee }_{ℋ}\perp \left(if\left({A}\in {\mathbf{C}}_{ℋ},{A},ℋ\right)\right)=ℋ$
7 4 6 dedth ${⊢}{A}\in {\mathbf{C}}_{ℋ}\to {A}{\vee }_{ℋ}\perp \left({A}\right)=ℋ$