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

Metamath Proof Explorer

Theorem opcon2b

Description: Orthocomplement contraposition law. ( negcon2 analog.) (Contributed by NM, 16-Jan-2012)

Ref Expression
Hypotheses opoccl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
opoccl.o ⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion opcon2b ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = ( ⊥ ‘ 𝑌 ) ↔ 𝑌 = ( ⊥ ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 opoccl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 opoccl.o ⊢ ⊥ = ( oc ‘ 𝐾 )
3 1 2 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
4 3 3adant2 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
5 1 2 opcon3b ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 = ( ⊥ ‘ 𝑌 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = ( ⊥ ‘ 𝑋 ) ) )
6 4 5 syld3an3 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = ( ⊥ ‘ 𝑌 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = ( ⊥ ‘ 𝑋 ) ) )
7 1 2 opococ ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
8 7 3adant2 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
9 8 eqeq1d ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = ( ⊥ ‘ 𝑋 ) ↔ 𝑌 = ( ⊥ ‘ 𝑋 ) ) )
10 6 9 bitrd ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = ( ⊥ ‘ 𝑌 ) ↔ 𝑌 = ( ⊥ ‘ 𝑋 ) ) )