Theorem ocvocv

Description: A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	ocvss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ocvss.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
Assertion	ocvocv	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → 𝑆 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑆 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ocvss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ocvss.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
3	1 2	ocvss	⊢ ( ⊥ ‘ 𝑆 ) ⊆ 𝑉
4	3	a1i	⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → ( ⊥ ‘ 𝑆 ) ⊆ 𝑉 )
5		simpr	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → 𝑆 ⊆ 𝑉 )
6	5	sselda	⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑉 )
7		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
8		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
9		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
10	1 7 8 9 2	ocvi	⊢ ( ( 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
11	10	ancoms	⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
12	11	adantll	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
13		simplll	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → 𝑊 ∈ PreHil )
14	4	sselda	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → 𝑦 ∈ 𝑉 )
15	6	adantr	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → 𝑥 ∈ 𝑉 )
16	8 7 1 9	iporthcom	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
17	13 14 15 16	syl3anc	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → ( ( 𝑦 ( ·𝑖 ‘ 𝑊 ) 𝑥 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
18	12 17	mpbid	⊢ ( ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ) → ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
19	18	ralrimiva	⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → ∀ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
20	1 7 8 9 2	elocv	⊢ ( 𝑥 ∈ ( ⊥ ‘ ( ⊥ ‘ 𝑆 ) ) ↔ ( ( ⊥ ‘ 𝑆 ) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ ∀ 𝑦 ∈ ( ⊥ ‘ 𝑆 ) ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
21	4 6 19 20	syl3anbrc	⊢ ( ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( ⊥ ‘ ( ⊥ ‘ 𝑆 ) ) )
22	21	ex	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → ( 𝑥 ∈ 𝑆 → 𝑥 ∈ ( ⊥ ‘ ( ⊥ ‘ 𝑆 ) ) ) )
23	22	ssrdv	⊢ ( ( 𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ) → 𝑆 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑆 ) ) )