Metamath Proof Explorer


Theorem osumi

Description: If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of Kalmbach p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth , although "the hard part" of this proof, chscl , requires no choice. (Contributed by NM, 28-Oct-1999) (Revised by Mario Carneiro, 19-May-2014) (New usage is discouraged.)

Ref Expression
Hypotheses osum.1
|- A e. CH
osum.2
|- B e. CH
Assertion osumi
|- ( A C_ ( _|_ ` B ) -> ( A +H B ) = ( A vH B ) )

Proof

Step Hyp Ref Expression
1 osum.1
 |-  A e. CH
2 osum.2
 |-  B e. CH
3 1 a1i
 |-  ( A C_ ( _|_ ` B ) -> A e. CH )
4 2 a1i
 |-  ( A C_ ( _|_ ` B ) -> B e. CH )
5 1 2 chsscon2i
 |-  ( A C_ ( _|_ ` B ) <-> B C_ ( _|_ ` A ) )
6 5 biimpi
 |-  ( A C_ ( _|_ ` B ) -> B C_ ( _|_ ` A ) )
7 3 4 6 chscl
 |-  ( A C_ ( _|_ ` B ) -> ( A +H B ) e. CH )
8 1 chshii
 |-  A e. SH
9 2 chshii
 |-  B e. SH
10 8 9 shjshseli
 |-  ( ( A +H B ) e. CH <-> ( A +H B ) = ( A vH B ) )
11 7 10 sylib
 |-  ( A C_ ( _|_ ` B ) -> ( A +H B ) = ( A vH B ) )