srgsummulcr

Description: A finite semiring sum multiplied by a constant, analogous to gsummulc1 . (Contributed by AV, 23-Aug-2019)

	Ref	Expression
Hypotheses	srgsummulcr.b	$⊢ B = {Base}_{R}$
	srgsummulcr.z	$⊢ \underset{˙}{0} = 0_{R}$
	srgsummulcr.p	$⊢ \underset{˙}{+} = +_{R}$
	srgsummulcr.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
	srgsummulcr.r	$⊢ φ \to R \in SRing$
	srgsummulcr.a	$⊢ φ \to A \in V$
	srgsummulcr.y	$⊢ φ \to Y \in B$
	srgsummulcr.x	$⊢ (φ \land k \in A) \to X \in B$
	srgsummulcr.n	$⊢ φ \to {finSupp}_{\underset{˙}{0}} ((k \in A ⟼ X))$
Assertion	srgsummulcr	$⊢ φ \to \underset{k \in A}{\sum_{R}} (X \underset{˙}{\cdot} Y) = (\underset{k \in A}{\sum_{R}}, X) \underset{˙}{\cdot} Y$

Step	Hyp	Ref	Expression
1		srgsummulcr.b	$⊢ B = {Base}_{R}$
2		srgsummulcr.z	$⊢ \underset{˙}{0} = 0_{R}$
3		srgsummulcr.p	$⊢ \underset{˙}{+} = +_{R}$
4		srgsummulcr.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
5		srgsummulcr.r	$⊢ φ \to R \in SRing$
6		srgsummulcr.a	$⊢ φ \to A \in V$
7		srgsummulcr.y	$⊢ φ \to Y \in B$
8		srgsummulcr.x	$⊢ (φ \land k \in A) \to X \in B$
9		srgsummulcr.n	$⊢ φ \to {finSupp}_{\underset{˙}{0}} ((k \in A ⟼ X))$
10		srgcmn	$⊢ R \in SRing \to R \in CMnd$
11	5 10	syl	$⊢ φ \to R \in CMnd$
12		srgmnd	$⊢ R \in SRing \to R \in Mnd$
13	5 12	syl	$⊢ φ \to R \in Mnd$
14	1 4	srgrmhm	$⊢ (R \in SRing \land Y \in B) \to (x \in B ⟼ x \underset{˙}{\cdot} Y) \in (R MndHom R)$
15	5 7 14	syl2anc	$⊢ φ \to (x \in B ⟼ x \underset{˙}{\cdot} Y) \in (R MndHom R)$
16		oveq1	$⊢ x = X \to x \underset{˙}{\cdot} Y = X \underset{˙}{\cdot} Y$
17		oveq1	$⊢ x = \underset{k \in A}{\sum_{R}} X \to x \underset{˙}{\cdot} Y = (\underset{k \in A}{\sum_{R}}, X) \underset{˙}{\cdot} Y$
18	1 2 11 13 6 15 8 9 16 17	gsummhm2	$⊢ φ \to \underset{k \in A}{\sum_{R}} (X \underset{˙}{\cdot} Y) = (\underset{k \in A}{\sum_{R}}, X) \underset{˙}{\cdot} Y$

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