Theorem ringidcld

Description: The unity element of a ring belongs to the base set of the ring, deduction version. (Contributed by SN, 16-Oct-2025)

	Ref	Expression
Hypotheses	ringidcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringidcl.u	⊢ 1 = ( 1r ‘ 𝑅 )
	ringidcld.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
Assertion	ringidcld	⊢ ( 𝜑 → 1 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ringidcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringidcl.u	⊢ 1 = ( 1r ‘ 𝑅 )
3		ringidcld.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
4	1 2	ringidcl	⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 )
5	3 4	syl	⊢ ( 𝜑 → 1 ∈ 𝐵 )