Theorem ltrncnvleN

Description: Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ltrnle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ltrnle.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnle.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnle.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrncnvleN	⊢ ( ( ( 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		simp3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
10	1 2 6	lautcnvle	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( ◡ 𝐹 ‘ 𝑋 ) ≤ ( ◡ 𝐹 ‘ 𝑌 ) ) )
11	5 8 9 10	syl21anc	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( ◡ 𝐹 ‘ 𝑋 ) ≤ ( ◡ 𝐹 ‘ 𝑌 ) ) )