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 ) )