Theorem mnringaddgd

Description: The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024) (Proof shortened by AV, 1-Nov-2024)

	Ref	Expression
Hypotheses	mnringaddgd.1	|- F = ( R MndRing M )
	mnringaddgd.2	|- A = ( Base ` M )
	mnringaddgd.3	|- V = ( R freeLMod A )
	mnringaddgd.4	|- ( ph -> R e. U )
	mnringaddgd.5	|- ( ph -> M e. W )
Assertion	mnringaddgd	|- ( ph -> ( +g ` V ) = ( +g ` F ) )

Proof

Step	Hyp	Ref	Expression
1		mnringaddgd.1	|- F = ( R MndRing M )
2		mnringaddgd.2	|- A = ( Base ` M )
3		mnringaddgd.3	|- V = ( R freeLMod A )
4		mnringaddgd.4	|- ( ph -> R e. U )
5		mnringaddgd.5	|- ( ph -> M e. W )
6		plusgid	|- +g = Slot ( +g ` ndx )
7		plusgndxnmulrndx	|- ( +g ` ndx ) =/= ( .r ` ndx )
8	1 6 7 2 3 4 5	mnringnmulrd	|- ( ph -> ( +g ` V ) = ( +g ` F ) )