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Metamath Proof Explorer

Theorem ltrn11

Description: One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ltrn1o.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
ltrn1o.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrn1o.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrn11 ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ltrn1o.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 ltrn1o.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 ltrn1o.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 simp1l ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ 𝑉 )
5 eqid ⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
6 2 5 3 ltrnlaut ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
7 6 3adant3 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
8 simp3l ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
9 simp3r ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
10 1 5 laut11 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )
11 4 7 8 9 10 syl22anc ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )