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Metamath Proof Explorer

Theorem oplecon1b

Description: Contraposition law for strict ordering in orthoposets. ( chsscon1 analog.) (Contributed by NM, 6-Nov-2011)

Ref Expression
Hypotheses opcon3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
opcon3.l ⊢ ≤ = ( le ‘ 𝐾 )
opcon3.o ⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion oplecon1b ( ( 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 1 3 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
5 4 3adant3 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
6 1 2 3 oplecon3b ⊢ ( ( 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 ) )