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 )