Theorem dochocsn

Description: The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015)

	Ref	Expression
Hypotheses	dochocsn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dochocsn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dochocsn.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	dochocsn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	dochocsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	dochocsn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dochocsn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion	dochocsn	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) = ( 𝑁 ‘ { 𝑋 } ) )

Proof

Step	Hyp	Ref	Expression
1		dochocsn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dochocsn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		dochocsn.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		dochocsn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		dochocsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6		dochocsn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		dochocsn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
8	7	snssd	⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
9	1 2 3 4 5 6 8	dochocsp	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( ⊥ ‘ { 𝑋 } ) )
10	9	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) )
11		eqid	⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
12	1 2 4 5 11	dihlsprn	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
13	6 7 12	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) )
14	1 11 3	dochoc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )
15	6 13 14	syl2anc	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( 𝑁 ‘ { 𝑋 } ) )
16	10 15	eqtr3d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ { 𝑋 } ) ) = ( 𝑁 ‘ { 𝑋 } ) )