Metamath Proof Explorer


Theorem pjpo

Description: Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999) (New usage is discouraged.)

Ref Expression
Assertion pjpo ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj𝐻 ) ‘ 𝐴 ) = ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 choccl ( 𝐻C → ( ⊥ ‘ 𝐻 ) ∈ C )
2 pjhcl ( ( ( ⊥ ‘ 𝐻 ) ∈ C𝐴 ∈ ℋ ) → ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
3 1 2 sylan ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
4 pjhcl ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj𝐻 ) ‘ 𝐴 ) ∈ ℋ )
5 ax-hvcom ( ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( proj𝐻 ) ‘ 𝐴 ) ∈ ℋ ) → ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) = ( ( ( proj𝐻 ) ‘ 𝐴 ) + ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
6 3 4 5 syl2anc ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) = ( ( ( proj𝐻 ) ‘ 𝐴 ) + ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
7 axpjpj ( ( 𝐻C𝐴 ∈ ℋ ) → 𝐴 = ( ( ( proj𝐻 ) ‘ 𝐴 ) + ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
8 6 7 eqtr4d ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) = 𝐴 )
9 simpr ( ( 𝐻C𝐴 ∈ ℋ ) → 𝐴 ∈ ℋ )
10 hvsubadd ( ( 𝐴 ∈ ℋ ∧ ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( proj𝐻 ) ‘ 𝐴 ) ∈ ℋ ) → ( ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( proj𝐻 ) ‘ 𝐴 ) ↔ ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) = 𝐴 ) )
11 9 3 4 10 syl3anc ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( proj𝐻 ) ‘ 𝐴 ) ↔ ( ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) = 𝐴 ) )
12 8 11 mpbird ( ( 𝐻C𝐴 ∈ ℋ ) → ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( proj𝐻 ) ‘ 𝐴 ) )
13 12 eqcomd ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj𝐻 ) ‘ 𝐴 ) = ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )