Theorem rmsuppfi

Description: The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019)

	Ref	Expression
Hypothesis	rmsuppfi.r	|- R = ( Base ` M )
Assertion	rmsuppfi	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` M ) ) e. Fin ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) e. Fin )

Proof

Step	Hyp	Ref	Expression
1		rmsuppfi.r	|- R = ( Base ` M )
2		simp3	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` M ) ) e. Fin ) -> ( A supp ( 0g ` M ) ) e. Fin )
3	1	rmsuppss	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) )
4	3	3adant3	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` M ) ) e. Fin ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) )
5		ssfi	|- ( ( ( A supp ( 0g ` M ) ) e. Fin /\ ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) C_ ( A supp ( 0g ` M ) ) ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) e. Fin )
6	2 4 5	syl2anc	|- ( ( ( M e. Ring /\ V e. X /\ C e. R ) /\ A e. ( R ^m V ) /\ ( A supp ( 0g ` M ) ) e. Fin ) -> ( ( v e. V |-> ( C ( .r ` M ) ( A ` v ) ) ) supp ( 0g ` M ) ) e. Fin )