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Metamath Proof Explorer

Theorem ltrnmw

Description: Property of lattice translation value. Remark below Lemma B in Crawley p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012) (Proof shortened by OpenAI, 25-Mar-2020)

Ref Expression
Hypotheses ltrnmw.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrnmw.m ⊢ ∧ = ( meet ‘ 𝐾 )
ltrnmw.z ⊢ 0 = ( 0. ‘ 𝐾 )
ltrnmw.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrnmw.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnmw.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrnmw ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )

Proof

Step Hyp Ref Expression
1 ltrnmw.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrnmw.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 ltrnmw.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 ltrnmw.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 ltrnmw.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 ltrnmw.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
9 1 2 3 4 5 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )
10 7 8 9 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )