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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 ∈ C ℋ ∧ B ∈ C ℋ ∧ A ⊆ ⊥ ⁡ B → A + ℋ B = A ∨ ℋ B

Proof

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