Theorem cvrle

Description: The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011)

	Ref	Expression
Hypotheses	cvrle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cvrle.l	⊢ ≤ = ( le ‘ 𝐾 )
	cvrle.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion	cvrle	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ≤ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		cvrle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvrle.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cvrle.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4		eqid	⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 )
5	1 4 3	cvrlt	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ( lt ‘ 𝐾 ) 𝑌 )
6	2 4	pltval	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( lt ‘ 𝐾 ) 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
7	6	simprbda	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) → 𝑋 ≤ 𝑌 )
8	5 7	syldan	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ≤ 𝑌 )