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Metamath Proof Explorer

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 ) )