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 ) ) )