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Metamath Proof Explorer

Theorem ocvz

Description: The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015)

Ref Expression
Hypotheses ocvz.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
ocvz.o ⊢ ⊥ = ( ocv ‘ 𝑊 )
ocvz.z ⊢ 0 = ( 0g ‘ 𝑊 )
Assertion ocvz ( 𝑊 ∈ PreHil → ( ⊥ ‘ { 0 } ) = 𝑉 )

Proof

Step Hyp Ref Expression
1 ocvz.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 ocvz.o ⊢ ⊥ = ( ocv ‘ 𝑊 )
3 ocvz.z ⊢ 0 = ( 0g ‘ 𝑊 )
4 phllmod ⊢ ( 𝑊 ∈ PreHil → 𝑊 ∈ LMod )
5 eqid ⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 )
6 3 5 lsp0 ⊢ ( 𝑊 ∈ LMod → ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) = { 0 } )
7 4 6 syl ⊢ ( 𝑊 ∈ PreHil → ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) = { 0 } )
8 7 fveq2d ⊢ ( 𝑊 ∈ PreHil → ( ⊥ ‘ ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) ) = ( ⊥ ‘ { 0 } ) )
9 0ss ⊢ ∅ ⊆ 𝑉
10 1 2 5 ocvlsp ⊢ ( ( 𝑊 ∈ PreHil ∧ ∅ ⊆ 𝑉 ) → ( ⊥ ‘ ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) ) = ( ⊥ ‘ ∅ ) )
11 9 10 mpan2 ⊢ ( 𝑊 ∈ PreHil → ( ⊥ ‘ ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) ) = ( ⊥ ‘ ∅ ) )
12 1 2 ocv0 ⊢ ( ⊥ ‘ ∅ ) = 𝑉
13 11 12 eqtrdi ⊢ ( 𝑊 ∈ PreHil → ( ⊥ ‘ ( ( LSpan ‘ 𝑊 ) ‘ ∅ ) ) = 𝑉 )
14 8 13 eqtr3d ⊢ ( 𝑊 ∈ PreHil → ( ⊥ ‘ { 0 } ) = 𝑉 )