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Metamath Proof Explorer

Theorem gsummulc2

Description: A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by AV, 10-Jul-2019) Remove unused hypothesis. (Revised by SN, 7-Mar-2025)

Ref Expression
Hypotheses gsummulc1.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
gsummulc1.z ⊢ 0 = ( 0g ‘ 𝑅 )
gsummulc1.t ⊢ · = ( .r ‘ 𝑅 )
gsummulc1.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
gsummulc1.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsummulc1.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
gsummulc1.x ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
gsummulc1.n ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
Assertion gsummulc2 ( 𝜑 → ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑌 · 𝑋 ) ) ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 gsummulc1.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 gsummulc1.z ⊢ 0 = ( 0g ‘ 𝑅 )
3 gsummulc1.t ⊢ · = ( .r ‘ 𝑅 )
4 gsummulc1.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
5 gsummulc1.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
6 gsummulc1.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 gsummulc1.x ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
8 gsummulc1.n ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
9 4 ringcmnd ⊢ ( 𝜑 → 𝑅 ∈ CMnd )
10 ringmnd ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
11 4 10 syl ⊢ ( 𝜑 → 𝑅 ∈ Mnd )
12 1 3 ringlghm ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
13 4 6 12 syl2anc ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
14 ghmmhm ⊢ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
15 13 14 syl ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
16 oveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝑌 · 𝑥 ) = ( 𝑌 · 𝑋 ) )
17 oveq2 ⊢ ( 𝑥 = ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) → ( 𝑌 · 𝑥 ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )
18 1 2 9 11 5 15 7 8 16 17 gsummhm2 ⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑌 · 𝑋 ) ) ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )