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Metamath Proof Explorer

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 ) )