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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 ( ( 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 ) ) )