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 C A A H norm proj H A = norm A

Proof

Step Hyp Ref Expression
1 pjhcl H C A proj H A
2 normcl proj H A norm proj H A
3 1 2 syl H C A norm proj H A
4 normcl A norm A
5 4 adantl H C A norm A
6 3 5 eqleltd H C A norm proj H A = norm A norm proj H A norm A ¬ norm proj H A < norm A
7 pjnorm H C A norm proj H A norm A
8 7 biantrurd H C A ¬ norm proj H A < norm A norm proj H A norm A ¬ norm proj H A < norm A
9 pjnel H C A ¬ A H norm proj H A < norm A
10 9 con1bid H C A ¬ norm proj H A < norm A A H
11 6 8 10 3bitr2rd H C A A H norm proj H A = norm A