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Metamath Proof Explorer

Theorem atcvrj2

Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012)

Ref Expression
Hypotheses atcvrj1x.l ⊢ ≤ = ( le ‘ 𝐾 )
atcvrj1x.j ⊢ ∨ = ( join ‘ 𝐾 )
atcvrj1x.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
atcvrj1x.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atcvrj2 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 atcvrj1x.l ⊢ ≤ = ( le ‘ 𝐾 )
2 atcvrj1x.j ⊢ ∨ = ( join ‘ 𝐾 )
3 atcvrj1x.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4 atcvrj1x.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 1 2 3 4 atcvrj2b ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑅 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ↔ 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) ) )
6 5 biimp3a ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → 𝑃 𝐶 ( 𝑄 ∨ 𝑅 ) )