Theorem pjoc2

Description: Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of Beran p. 111. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjoc2	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 0ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ⊥ ‘ 𝐻 ) = ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) )
2	1	eleq2d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ 𝐴 ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ) )
3		fveq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) )
4	3	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) )
5	4	eqeq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 0ℎ ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) = 0ℎ ) )
6	2 5	bibi12d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 0ℎ ) ↔ ( 𝐴 ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) = 0ℎ ) ) )
7		eleq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ) )
8		fveqeq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) = 0ℎ ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ ) )
9	7 8	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) = 0ℎ ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ ) ) )
10		h0elch	⊢ 0ℋ ∈ Cℋ
11	10	elimel	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ∈ Cℋ
12		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
13	11 12	pjoc2i	⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ( ⊥ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = 0ℎ )
14	6 9 13	dedth2h	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 0ℎ ) )