Theorem pjoc1

Description: Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjoc1	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( 𝐴 ∈ 𝐻 ↔ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
2		2fveq3	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) = ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) )
3	2	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) )
4	3	eqeq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0ℎ ) )
5	1 4	bibi12d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ) ↔ ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0ℎ ) ) )
6		eleq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
7		fveqeq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0ℎ ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ ) )
8	6 7	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0ℎ ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ ) ) )
9		ifchhv	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ∈ Cℋ
10		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
11	9 10	pjoc1i	⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ )
12	5 8 11	dedth2h	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0ℎ ) )