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Metamath Proof Explorer

Theorem leat2

Description: A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013)

Ref Expression
Hypotheses leatom.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
leatom.l ⊢ ≤ = ( le ‘ 𝐾 )
leatom.z ⊢ 0 = ( 0. ‘ 𝐾 )
leatom.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion leat2 ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃 ) ) → 𝑋 = 𝑃 )

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 leatb ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ) )
6 orcom ⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) )
7 neor ⊢ ( ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) )
8 6 7 bitri ⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) )
9 5 8 bitrdi ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) )
10 9 biimpd ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 → ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) )
11 10 com23 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≠ 0 → ( 𝑋 ≤ 𝑃 → 𝑋 = 𝑃 ) ) )
12 11 imp32 ⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃 ) ) → 𝑋 = 𝑃 )