Step |
Hyp |
Ref |
Expression |
1 |
|
gsummulgc1.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
gsummulgc1.t |
⊢ · = ( .g ‘ 𝑀 ) |
3 |
|
gsummulgc1.r |
⊢ ( 𝜑 → 𝑀 ∈ Grp ) |
4 |
|
gsummulgc1.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
gsummulgc1.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
gsummulgc1.x |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ ℤ ) |
7 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
8 |
|
zring0 |
⊢ 0 = ( 0g ‘ ℤring ) |
9 |
|
zringring |
⊢ ℤring ∈ Ring |
10 |
|
ringcmn |
⊢ ( ℤring ∈ Ring → ℤring ∈ CMnd ) |
11 |
9 10
|
mp1i |
⊢ ( 𝜑 → ℤring ∈ CMnd ) |
12 |
3
|
grpmndd |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
13 |
|
eqid |
⊢ ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) = ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝑌 ) ) |
14 |
2 13 1
|
mulgghm2 |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 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 |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) = ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) |
19 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
20 |
18 4 6 19
|
fsuppmptdm |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 ) |
21 |
|
oveq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 · 𝑌 ) = ( 𝑋 · 𝑌 ) ) |
22 |
|
oveq1 |
⊢ ( 𝑥 = ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) → ( 𝑥 · 𝑌 ) = ( ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) ) |
23 |
7 8 11 12 4 17 6 20 21 22
|
gsummhm2 |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) ) |
24 |
4 6
|
gsumzrsum |
⊢ ( 𝜑 → ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = Σ 𝑘 ∈ 𝐴 𝑋 ) |
25 |
24
|
oveq1d |
⊢ ( 𝜑 → ( ( ℤring Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) · 𝑌 ) = ( Σ 𝑘 ∈ 𝐴 𝑋 · 𝑌 ) ) |
26 |
23 25
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑋 · 𝑌 ) ) ) = ( Σ 𝑘 ∈ 𝐴 𝑋 · 𝑌 ) ) |