Theorem opoccl

Description: Closure of orthocomplement operation. ( choccl analog.) (Contributed by NM, 20-Oct-2011)

	Ref	Expression
Hypotheses	opoccl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opoccl.o	⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion	opoccl	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		opoccl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		opoccl.o	⊢ ⊥ = ( oc ‘ 𝐾 )
3		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
5		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
6		eqid	⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
7		eqid	⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
8	1 3 2 4 5 6 7	oposlem	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ( join ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 0. ‘ 𝐾 ) ) )
9	8	3anidm23	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ( join ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 0. ‘ 𝐾 ) ) )
10	9	simp1d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) )
11	10	simp1d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )