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Metamath Proof Explorer

Theorem ltrnle

Description: Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ltrnle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ltrnle.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrnle.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnle.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrnle ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 ltrnle.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ltrnle.l ⊢ ≤ = ( le ‘ 𝐾 )
3 ltrnle.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrnle.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 simp1l ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ 𝑉 )
6 eqid ⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
7 3 6 4 ltrnlaut ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
8 7 3adant3 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
9 simp3l ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
10 simp3r ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
11 1 2 6 lautle ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )
12 5 8 9 10 11 syl22anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )