Theorem oplecon3

Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011)

	Ref	Expression
Hypotheses	opcon3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opcon3.l	⊢ ≤ = ( le ‘ 𝐾 )
	opcon3.o	⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion	oplecon3	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( ⊥ ‘ 𝑌 ) ≤ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		opcon3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		opcon3.l	⊢ ≤ = ( le ‘ 𝐾 )
3		opcon3.o	⊢ ⊥ = ( oc ‘ 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 2 3 4 5 6 7	oposlem	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ≤ 𝑌 → ( ⊥ ‘ 𝑌 ) ≤ ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ( join ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 0. ‘ 𝐾 ) ) )
9	8	simp1d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ≤ 𝑌 → ( ⊥ ‘ 𝑌 ) ≤ ( ⊥ ‘ 𝑋 ) ) ) )
10	9	simp3d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( ⊥ ‘ 𝑌 ) ≤ ( ⊥ ‘ 𝑋 ) ) )