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 ( ( 𝐻C𝐴 ∈ ℋ ) → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( ( proj𝐻 ) ‘ 𝐴 ) = 0 ) )

Proof

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