Theorem pjpo

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

	Ref	Expression
Assertion	pjpo	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) ∈ Cℋ )
2		pjhcl	⊢ ( ( ( ⊥ ‘ 𝐻 ) ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
3	1 2	sylan	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
4		pjhcl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )
5		ax-hvcom	⊢ ( ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
6	3 4 5	syl2anc	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
7		axpjpj	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
8	6 7	eqtr4d	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 𝐴 )
9		simpr	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 𝐴 ∈ ℋ )
10		hvsubadd	⊢ ( ( 𝐴 ∈ ℋ ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ ) → ( ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ↔ ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 𝐴 ) )
11	9 3 4 10	syl3anc	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ↔ ( ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = 𝐴 ) )
12	8 11	mpbird	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) )
13	12	eqcomd	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )