Step |
Hyp |
Ref |
Expression |
1 |
|
mgpsumunsn.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
2 |
|
mgpsumunsn.t |
⊢ · = ( .r ‘ 𝑅 ) |
3 |
|
mgpsumunsn.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
4 |
|
mgpsumunsn.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
5 |
|
mgpsumunsn.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑁 ) |
6 |
|
mgpsumunsn.a |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑁 ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
7 |
|
mgpsumz.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
8 |
|
mgpsumz.0 |
⊢ ( 𝑘 = 𝐼 → 𝐴 = 0 ) |
9 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
10 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
11 |
9 10
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Mnd ) |
12 |
3 11
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
14 |
13 7
|
mndidcl |
⊢ ( 𝑅 ∈ Mnd → 0 ∈ ( Base ‘ 𝑅 ) ) |
15 |
12 14
|
syl |
⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝑅 ) ) |
16 |
1 2 3 4 5 6 15 8
|
mgpsumunsn |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ 𝑁 ↦ 𝐴 ) ) = ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) ) |
17 |
3 9
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
18 |
1 13
|
mgpbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 ) |
19 |
1
|
crngmgp |
⊢ ( 𝑅 ∈ CRing → 𝑀 ∈ CMnd ) |
20 |
3 19
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ CMnd ) |
21 |
|
diffi |
⊢ ( 𝑁 ∈ Fin → ( 𝑁 ∖ { 𝐼 } ) ∈ Fin ) |
22 |
4 21
|
syl |
⊢ ( 𝜑 → ( 𝑁 ∖ { 𝐼 } ) ∈ Fin ) |
23 |
|
eldifi |
⊢ ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) → 𝑘 ∈ 𝑁 ) |
24 |
23 6
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
25 |
24
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
26 |
18 20 22 25
|
gsummptcl |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) ∈ ( Base ‘ 𝑅 ) ) |
27 |
13 2 7
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) = 0 ) |
28 |
17 26 27
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) = 0 ) |
29 |
16 28
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ 𝑁 ↦ 𝐴 ) ) = 0 ) |