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Metamath Proof Explorer

Theorem ltrncoelN

Description: Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel uses. (Contributed by NM, 1-May-2013) (New usage is discouraged.)

Ref Expression
Hypotheses ltrnel.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrnel.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrnel.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnel.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncoelN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 ltrnel.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrnel.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrnel.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrnel.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
7 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
8 7 3adant2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
9 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ 𝑊 ) )
10 5 6 8 9 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ 𝑊 ) )