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 = ( 0g𝑅 )
gsummulc1.t · = ( .r𝑅 )
gsummulc1.r ( 𝜑𝑅 ∈ Ring )
gsummulc1.a ( 𝜑𝐴𝑉 )
gsummulc1.y ( 𝜑𝑌𝐵 )
gsummulc1.x ( ( 𝜑𝑘𝐴 ) → 𝑋𝐵 )
gsummulc1.n ( 𝜑 → ( 𝑘𝐴𝑋 ) finSupp 0 )
Assertion gsummulc2 ( 𝜑 → ( 𝑅 Σg ( 𝑘𝐴 ↦ ( 𝑌 · 𝑋 ) ) ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘𝐴𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 gsummulc1.b 𝐵 = ( Base ‘ 𝑅 )
2 gsummulc1.z 0 = ( 0g𝑅 )
3 gsummulc1.t · = ( .r𝑅 )
4 gsummulc1.r ( 𝜑𝑅 ∈ Ring )
5 gsummulc1.a ( 𝜑𝐴𝑉 )
6 gsummulc1.y ( 𝜑𝑌𝐵 )
7 gsummulc1.x ( ( 𝜑𝑘𝐴 ) → 𝑋𝐵 )
8 gsummulc1.n ( 𝜑 → ( 𝑘𝐴𝑋 ) finSupp 0 )
9 4 ringcmnd ( 𝜑𝑅 ∈ CMnd )
10 ringmnd ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
11 4 10 syl ( 𝜑𝑅 ∈ Mnd )
12 1 3 ringlghm ( ( 𝑅 ∈ Ring ∧ 𝑌𝐵 ) → ( 𝑥𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
13 4 6 12 syl2anc ( 𝜑 → ( 𝑥𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
14 ghmmhm ( ( 𝑥𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) → ( 𝑥𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
15 13 14 syl ( 𝜑 → ( 𝑥𝐵 ↦ ( 𝑌 · 𝑥 ) ) ∈ ( 𝑅 MndHom 𝑅 ) )
16 oveq2 ( 𝑥 = 𝑋 → ( 𝑌 · 𝑥 ) = ( 𝑌 · 𝑋 ) )
17 oveq2 ( 𝑥 = ( 𝑅 Σg ( 𝑘𝐴𝑋 ) ) → ( 𝑌 · 𝑥 ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘𝐴𝑋 ) ) ) )
18 1 2 9 11 5 15 7 8 16 17 gsummhm2 ( 𝜑 → ( 𝑅 Σg ( 𝑘𝐴 ↦ ( 𝑌 · 𝑋 ) ) ) = ( 𝑌 · ( 𝑅 Σg ( 𝑘𝐴𝑋 ) ) ) )