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 ∈ Cℋ
	osum.2	⊢ 𝐵 ∈ Cℋ
Assertion	osumi	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		osum.1	⊢ 𝐴 ∈ Cℋ
2		osum.2	⊢ 𝐵 ∈ Cℋ
3	1	a1i	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → 𝐴 ∈ Cℋ )
4	2	a1i	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → 𝐵 ∈ Cℋ )
5	1 2	chsscon2i	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) )
6	5	biimpi	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) )
7	3 4 6	chscl	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) ∈ Cℋ )
8	1	chshii	⊢ 𝐴 ∈ Sℋ
9	2	chshii	⊢ 𝐵 ∈ Sℋ
10	8 9	shjshseli	⊢ ( ( 𝐴 +ℋ 𝐵 ) ∈ Cℋ ↔ ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )
11	7 10	sylib	⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )