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Metamath Proof Explorer

Theorem lhplt

Description: An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012)

Ref Expression
Hypotheses lhplt.l ⊢ ≤ = ( le ‘ 𝐾 )
lhplt.s ⊢ < = ( lt ‘ 𝐾 )
lhplt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lhplt.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion lhplt ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑃 < 𝑊 )

Proof

Step Hyp Ref Expression
1 lhplt.l ⊢ ≤ = ( le ‘ 𝐾 )
2 lhplt.s ⊢ < = ( lt ‘ 𝐾 )
3 lhplt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 lhplt.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
6 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 4 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
9 8 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
10 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
11 eqid ⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
12 10 11 4 lhp1cvr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) )
14 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑃 ≤ 𝑊 )
15 7 1 2 10 11 3 1cvratlt ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ∧ 𝑃 ≤ 𝑊 ) ) → 𝑃 < 𝑊 )
16 5 6 9 13 14 15 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) → 𝑃 < 𝑊 )