Theorem unitcl

Description: A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014)

	Ref	Expression
Hypotheses	unitcl.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	unitcl.2	⊢ 𝑈 = ( Unit ‘ 𝑅 )
Assertion	unitcl	⊢ ( 𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		unitcl.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		unitcl.2	⊢ 𝑈 = ( Unit ‘ 𝑅 )
3		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
4		eqid	⊢ ( ∥r ‘ 𝑅 ) = ( ∥r ‘ 𝑅 )
5		eqid	⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
6		eqid	⊢ ( ∥r ‘ ( oppr ‘ 𝑅 ) ) = ( ∥r ‘ ( oppr ‘ 𝑅 ) )
7	2 3 4 5 6	isunit	⊢ ( 𝑋 ∈ 𝑈 ↔ ( 𝑋 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝑋 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) )
8	7	simplbi	⊢ ( 𝑋 ∈ 𝑈 → 𝑋 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) )
9	1 4	dvdsrcl	⊢ ( 𝑋 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) → 𝑋 ∈ 𝐵 )
10	8 9	syl	⊢ ( 𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵 )