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Metamath Proof Explorer

Theorem opnoncon

Description: Law of contradiction for orthoposets. ( chocin analog.) (Contributed by NM, 13-Sep-2011)

Ref Expression
Hypotheses opnoncon.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
opnoncon.o ⊢ ⊥ = ( oc ‘ 𝐾 )
opnoncon.m ⊢ ∧ = ( meet ‘ 𝐾 )
opnoncon.z ⊢ 0 = ( 0. ‘ 𝐾 )
Assertion opnoncon ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ ( ⊥ ‘ 𝑋 ) ) = 0 )

Proof

Step Hyp Ref Expression
1 opnoncon.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 opnoncon.o ⊢ ⊥ = ( oc ‘ 𝐾 )
3 opnoncon.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 opnoncon.z ⊢ 0 = ( 0. ‘ 𝐾 )
5 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
8 1 5 2 6 3 4 7 oposlem ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ( join ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑋 ∧ ( ⊥ ‘ 𝑋 ) ) = 0 ) )
9 8 3anidm23 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ( join ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑋 ∧ ( ⊥ ‘ 𝑋 ) ) = 0 ) )
10 9 simp3d ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ ( ⊥ ‘ 𝑋 ) ) = 0 )