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Metamath Proof Explorer

Theorem pjnorm2

Description: A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch yield Theorem 26.3 of Halmos p. 44. (Contributed by NM, 7-Apr-2001) (New usage is discouraged.)

Ref Expression
Assertion pjnorm2 ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 pjhcl ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )
2 normcl ⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℝ )
3 1 2 syl ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℝ )
4 normcl ⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℝ )
5 4 adantl ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ 𝐴 ) ∈ ℝ )
6 3 5 eqleltd ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ↔ ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ∧ ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) )
7 pjnorm ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) )
8 7 biantrurd ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ↔ ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ∧ ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) )
9 pjnel ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) )
10 9 con1bid ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ↔ 𝐴 ∈ 𝐻 ) )
11 6 8 10 3bitr2rd ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) )