Theorem pjch

Description: Projection of a vector in the projection subspace. Lemma 4.4(ii) of Beran p. 111. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjch	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( 𝐴 ∈ 𝐻 ↔ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
2		fveq2	⊢ ( 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 ) = 𝐴 ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) )
5	1 4	bibi12d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) ↔ ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) ) )
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		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
8		id	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
9	7 8	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
10	6 9	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
11		ifchhv	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ∈ Cℋ
12		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
13	11 12	pjchi	⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
14	5 10 13	dedth2h	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) )