Theorem osum

Description: If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of Kalmbach p. 67. (Contributed by NM, 31-Oct-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	osum	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ 𝐵 ) ) )
2		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) )
3		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) )
4	2 3	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) )
5	1 4	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ 𝐵 ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) ) )
6		fveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ⊥ ‘ 𝐵 ) = ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
7	6	sseq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ 𝐵 ) ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
8		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
9		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
10	8 9	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) )
11	7 10	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ 𝐵 ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ 𝐵 ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ 𝐵 ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) ) ) )
12		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
13		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
14	12 13	osumi	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊆ ( ⊥ ‘ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) +ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
15	5 11 14	dedth2h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) )
16	15	3impia	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )