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Metamath Proof Explorer

Theorem 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 ⊢ 0 = ( 0g ‘ 𝑅 )
srgsummulcr.p ⊢ + = ( +g ‘ 𝑅 )
srgsummulcr.t ⊢ · = ( .r ‘ 𝑅 )
srgsummulcr.r ⊢ ( 𝜑 → 𝑅 ∈ SRing )
srgsummulcr.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
srgsummulcr.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
srgsummulcr.x ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
srgsummulcr.n ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
Assertion srgsummulcr ( 𝜑 → ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) )

Proof

Step Hyp Ref Expression
1 srgsummulcr.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 srgsummulcr.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 srgsummulcr.p ⊢ + = ( +g ‘ 𝑅 )
4 srgsummulcr.t ⊢ · = ( .r ‘ 𝑅 )
5 srgsummulcr.r ⊢ ( 𝜑 → 𝑅 ∈ SRing )
6 srgsummulcr.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
7 srgsummulcr.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
8 srgsummulcr.x ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
9 srgsummulcr.n ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
10 srgcmn ⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ CMnd )
11 5 10 syl ⊢ ( 𝜑 → 𝑅 ∈ CMnd )
12 srgmnd ⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ Mnd )
13 5 12 syl ⊢ ( 𝜑 → 𝑅 ∈ Mnd )
14 1 4 srgrmhm ⊢ ( ( 𝑅 ∈ SRing ∧ 𝑌 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 · 𝑌 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
15 5 7 14 syl2anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 · 𝑌 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
16 oveq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 · 𝑌 ) = ( 𝑋 · 𝑌 ) )
17 oveq1 ⊢ ( 𝑥 = ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) → ( 𝑥 · 𝑌 ) = ( ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) )
18 1 2 11 13 6 15 8 9 16 17 gsummhm2 ⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) )