Theorem grpinvcld

Description: A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025)

	Ref	Expression
Hypotheses	grpinvcld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpinvcld.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
	grpinvcld.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	grpinvcld.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	grpinvcld	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpinvcld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpinvcld.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
3		grpinvcld.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
4		grpinvcld.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5	1 2	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )
6	3 4 5	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )