Theorem zlmbas

Description: Base set of a ZZ -module. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 3-Nov-2024)

	Ref	Expression
Hypotheses	zlmbas.w	|- W = ( ZMod ` G )
	zlmbas.2	|- B = ( Base ` G )
Assertion	zlmbas	|- B = ( Base ` W )

Proof

Step	Hyp	Ref	Expression
1		zlmbas.w	|- W = ( ZMod ` G )
2		zlmbas.2	|- B = ( Base ` G )
3		baseid	|- Base = Slot ( Base ` ndx )
4		scandxnbasendx	|- ( Scalar ` ndx ) =/= ( Base ` ndx )
5	4	necomi	|- ( Base ` ndx ) =/= ( Scalar ` ndx )
6		vscandxnbasendx	|- ( .s ` ndx ) =/= ( Base ` ndx )
7	6	necomi	|- ( Base ` ndx ) =/= ( .s ` ndx )
8	1 3 5 7	zlmlem	|- ( Base ` G ) = ( Base ` W )
9	2 8	eqtri	|- B = ( Base ` W )