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Metamath Proof Explorer

Theorem ltrn2ateq

Description: Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012)

Ref Expression
Hypotheses ltrn2eq.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrn2eq.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrn2eq.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrn2eq.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrn2ateq ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )

Proof

Step Hyp Ref Expression
1 ltrn2eq.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrn2eq.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrn2eq.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrn2eq.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6 5 1 2 3 4 ltrnideq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
7 6 3adant3r3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
8 5 1 2 3 4 ltrnideq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )
9 8 3adant3r2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )
10 7 9 bitr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )