Theorem matgrp

Description: The matrix ring is a group. (Contributed by AV, 21-Dec-2019)

	Ref	Expression
Hypothesis	matlmod.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
Assertion	matgrp	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		matlmod.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2	1	matlmod	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ LMod )
3		lmodgrp	⊢ ( 𝐴 ∈ LMod → 𝐴 ∈ Grp )
4	2 3	syl	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Grp )