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Metamath Proof Explorer

Theorem 2atmat0

Description: The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012)

Ref Expression
Hypotheses 2atmatz.j ⊢ ∨ = ( join ‘ 𝐾 )
2atmatz.m ⊢ ∧ = ( meet ‘ 𝐾 )
2atmatz.z ⊢ 0 = ( 0. ‘ 𝐾 )
2atmatz.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 2atmat0 ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) = 0 ) )

Proof

Step Hyp Ref Expression
1 2atmatz.j ⊢ ∨ = ( join ‘ 𝐾 )
2 2atmatz.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 2atmatz.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 2atmatz.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) )
6 simpr1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → 𝑅 ∈ 𝐴 )
7 simpr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → 𝑆 ∈ 𝐴 )
8 7 orcd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( 𝑆 ∈ 𝐴 ∨ 𝑆 = 0 ) )
9 simpr3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) )
10 1 2 3 4 2at0mat0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∨ 𝑆 = 0 ) ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) = 0 ) )
11 5 6 8 9 10 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ≠ ( 𝑅 ∨ 𝑆 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ∨ 𝑆 ) ) = 0 ) )