Metamath Proof Explorer


Theorem gsummulgc2

Description: A finite group sum multiplied by a constant. (Contributed by Thierry Arnoux, 5-Oct-2025)

Ref Expression
Hypotheses gsummulgc1.b 𝐵 = ( Base ‘ 𝑀 )
gsummulgc1.t · = ( .g𝑀 )
gsummulgc1.r ( 𝜑𝑀 ∈ Grp )
gsummulgc1.a ( 𝜑𝐴 ∈ Fin )
gsummulgc1.y ( 𝜑𝑌𝐵 )
gsummulgc1.x ( ( 𝜑𝑘𝐴 ) → 𝑋 ∈ ℤ )
Assertion gsummulgc2 ( 𝜑 → ( 𝑀 Σg ( 𝑘𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( Σ 𝑘𝐴 𝑋 · 𝑌 ) )

Proof

Step Hyp Ref Expression
1 gsummulgc1.b 𝐵 = ( Base ‘ 𝑀 )
2 gsummulgc1.t · = ( .g𝑀 )
3 gsummulgc1.r ( 𝜑𝑀 ∈ Grp )
4 gsummulgc1.a ( 𝜑𝐴 ∈ Fin )
5 gsummulgc1.y ( 𝜑𝑌𝐵 )
6 gsummulgc1.x ( ( 𝜑𝑘𝐴 ) → 𝑋 ∈ ℤ )
7 zringbas ℤ = ( Base ‘ ℤring )
8 zring0 0 = ( 0g ‘ ℤring )
9 zringring ring ∈ Ring
10 ringcmn ( ℤring ∈ Ring → ℤring ∈ CMnd )
11 9 10 mp1i ( 𝜑 → ℤring ∈ CMnd )
12 3 grpmndd ( 𝜑𝑀 ∈ Mnd )
13 eqid ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) = ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) )
14 2 13 1 mulgghm2 ( ( 𝑀 ∈ Grp ∧ 𝑌𝐵 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) ∈ ( ℤring GrpHom 𝑀 ) )
15 3 5 14 syl2anc ( 𝜑 → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) ∈ ( ℤring GrpHom 𝑀 ) )
16 ghmmhm ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) ∈ ( ℤring GrpHom 𝑀 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) ∈ ( ℤring MndHom 𝑀 ) )
17 15 16 syl ( 𝜑 → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) ∈ ( ℤring MndHom 𝑀 ) )
18 eqid ( 𝑘𝐴𝑋 ) = ( 𝑘𝐴𝑋 )
19 0zd ( 𝜑 → 0 ∈ ℤ )
20 18 4 6 19 fsuppmptdm ( 𝜑 → ( 𝑘𝐴𝑋 ) finSupp 0 )
21 oveq1 ( 𝑥 = 𝑋 → ( 𝑥 · 𝑌 ) = ( 𝑋 · 𝑌 ) )
22 oveq1 ( 𝑥 = ( ℤring Σg ( 𝑘𝐴𝑋 ) ) → ( 𝑥 · 𝑌 ) = ( ( ℤring Σg ( 𝑘𝐴𝑋 ) ) · 𝑌 ) )
23 7 8 11 12 4 17 6 20 21 22 gsummhm2 ( 𝜑 → ( 𝑀 Σg ( 𝑘𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( ( ℤring Σg ( 𝑘𝐴𝑋 ) ) · 𝑌 ) )
24 4 6 gsumzrsum ( 𝜑 → ( ℤring Σg ( 𝑘𝐴𝑋 ) ) = Σ 𝑘𝐴 𝑋 )
25 24 oveq1d ( 𝜑 → ( ( ℤring Σg ( 𝑘𝐴𝑋 ) ) · 𝑌 ) = ( Σ 𝑘𝐴 𝑋 · 𝑌 ) )
26 23 25 eqtrd ( 𝜑 → ( 𝑀 Σg ( 𝑘𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( Σ 𝑘𝐴 𝑋 · 𝑌 ) )