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 ) )