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Metamath Proof Explorer

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 ) ) )