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 ( ( 𝐻C𝐴 ∈ ℋ ) → ( 𝐴𝐻 ↔ ( norm ‘ ( ( proj𝐻 ) ‘ 𝐴 ) ) = ( norm𝐴 ) ) )

Proof

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