Metamath Proof Explorer


Theorem leat3

Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012)

Ref Expression
Hypotheses leatom.b 𝐵 = ( Base ‘ 𝐾 )
leatom.l = ( le ‘ 𝐾 )
leatom.z 0 = ( 0. ‘ 𝐾 )
leatom.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion leat3 ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → ( 𝑋𝐴𝑋 = 0 ) )

Proof

Step Hyp Ref Expression
1 leatom.b 𝐵 = ( Base ‘ 𝐾 )
2 leatom.l = ( le ‘ 𝐾 )
3 leatom.z 0 = ( 0. ‘ 𝐾 )
4 leatom.a 𝐴 = ( Atoms ‘ 𝐾 )
5 1 2 3 4 leat ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → ( 𝑋 = 𝑃𝑋 = 0 ) )
6 simpl3 ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → 𝑃𝐴 )
7 eleq1a ( 𝑃𝐴 → ( 𝑋 = 𝑃𝑋𝐴 ) )
8 6 7 syl ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → ( 𝑋 = 𝑃𝑋𝐴 ) )
9 8 orim1d ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → ( ( 𝑋 = 𝑃𝑋 = 0 ) → ( 𝑋𝐴𝑋 = 0 ) ) )
10 5 9 mpd ( ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴 ) ∧ 𝑋 𝑃 ) → ( 𝑋𝐴𝑋 = 0 ) )