Theorem opltcon1b

Description: Contraposition law for strict ordering in orthoposets. ( chpsscon1 analog.) (Contributed by NM, 5-Nov-2011)

	Ref	Expression
Hypotheses	opltcon3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opltcon3.s	⊢ < = ( lt ‘ 𝐾 )
	opltcon3.o	⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion	opltcon1b	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) < 𝑌 ↔ ( ⊥ ‘ 𝑌 ) < 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		opltcon3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		opltcon3.s	⊢ < = ( lt ‘ 𝐾 )
3		opltcon3.o	⊢ ⊥ = ( oc ‘ 𝐾 )
4	1 3	opoccl	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
5	4	3adant3	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
6	1 2 3	opltcon3b	⊢ ( ( 𝐾 ∈ OP ∧ ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) < 𝑌 ↔ ( ⊥ ‘ 𝑌 ) < ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) )
7	5 6	syld3an2	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) < 𝑌 ↔ ( ⊥ ‘ 𝑌 ) < ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) )
8	1 3	opococ	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
9	8	3adant3	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
10	9	breq2d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑌 ) < ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ↔ ( ⊥ ‘ 𝑌 ) < 𝑋 ) )
11	7 10	bitrd	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) < 𝑌 ↔ ( ⊥ ‘ 𝑌 ) < 𝑋 ) )