Theorem zlmmulr

Description: Ring operation of a ZZ -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	zlmbas.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
	zlmmulr.2	⊢ · = ( .r ‘ 𝐺 )
Assertion	zlmmulr	⊢ · = ( .r ‘ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	⊢ 𝑊 = ( ℤMod ‘ 𝐺 )
2		zlmmulr.2	⊢ · = ( .r ‘ 𝐺 )
3		df-mulr	⊢ .r = Slot 3
4		3nn	⊢ 3 ∈ ℕ
5		3lt5	⊢ 3 < 5
6	1 3 4 5	zlmlem	⊢ ( .r ‘ 𝐺 ) = ( .r ‘ 𝑊 )
7	2 6	eqtri	⊢ · = ( .r ‘ 𝑊 )