Theorem ltrnval1

Description: Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ltrnval1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ltrnval1.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnval1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnval1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnval1	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		ltrnval1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ltrnval1.l	⊢ ≤ = ( le ‘ 𝐾 )
3		ltrnval1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrnval1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		eqid	⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
6	3 5 4	ltrnldil	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
7	6	3adant3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
8	1 2 3 5	ldilval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )
9	7 8	syld3an2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )