This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem opococ

Description: Double negative law for orthoposets. ( ococ analog.) (Contributed by NM, 13-Sep-2011)

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

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 simp2d ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )